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Computers As a Novel Mathematical Reality: III. Mersenne Numbers and Sums of Divisors

  • ALGORITHMIC MATHEMATICS AND MATHEMATICAL MODELING
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Abstract

Nowhere in mathematics is the progress resulting from the advent of computers as apparent as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behavior more than 2500 years ago. A description of the trajectories of this function—perfect numbers, amicable numbers, sociable numbers, and the like—constitute the contents of several problems stated over 2500 years ago, which still seem completely impenetrable. The theorem of Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced by hand, and since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs daily than were constructed by human beings over several millennia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson.

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Notes

  1. I deliberately do not indicate the exact value: today, hundreds of thousands of new such pairs are discovered every day, and so the exact number of known pairs is likely to change simply over the period of editorial preparation of this article.

  2. Stewart [327] put the question differently, “do dice play God”?

  3. In fact, Thomas Nicely discovered the FDIV bug in 1994 while calculating the Brun constant, the sum of reciprocals of simple twins.

  4. It is not trivial to do this even in compendium format [311, 312].

  5. I cannot resist quoting the following illustrative fragment from Robert Juricevic’s review of this book: “It seems that we will only begin to seriously understand the sequence of prime numbers when we are freely able to work with prime numbers which are at least 1 million digits in length. It would certainly be fantastic to discover a trick to do arithmetic with such huge fundamental building numbers without the aid of a computing machine. It would also be nice to be able to fly without the aid of a flying machine. Plainly, the computer is an indispensable tool to the theoretical mathematician studying the sequence of prime numbers, as well as to the mathematician applying prime number theory in industry”—“If only I could have such feathers, and such wings ….”

  6. In fact, as for [89, 140], we, unfortunately, used the previous edition. The second edition has been significantly expanded and updated precisely in the part relating to the factorization of Mersenne numbers and the role of computers in this.

  7. “Multiply 2 071 723\(~ \cdot \)5 363 222 357 by hand. Feel the joy.”

  8. “There is much pleasure to be gained from useless knowledge.”

  9. Constance Reed is the sister of Julia Robinson.

  10. Formally, the book Trattato de Numeri Perfetti [57] containing these results was published in 1603. However, its first sentence is “Nel trattato dè numeri perfetti, che giàsino dell’anno 1588 composi, ….” On p. 40, a table of all prime numbers \(p < 750\) is reproduced. Since \({{727}^{2}} = 528{\kern 1pt} 529 > {{M}_{{19}}}\), without even particularly delving in the text, it is clear how exactly Cataldi acted. He tested as possible divisors of \({{M}_{{13}}}\), \({{M}_{{17}}}\), and \({{M}_{{19}}}\) all primes not exceeding the integer part of their square roots—“sua prossima radice quadra.” It is beyond belief that Cataldi could extend this kind of direct calculation to \({{2897}^{2}} = 8\,392\,609 > {{M}_{{23}}}\).

  11. In accordance with the custom of the time, he says “any prescribed amount.”

  12. Both of these text fragments are reproduced in full in Latin in an article by Walter William Rose Ball [310], where they are, of course, somewhat easier to read than in the original 17th-century edition. The last sentence of this paragraph in the original reads as follows: “agnoscere num dati numeri 15, aut 20 caracteribus constantes, sint primi necne, cùm nequidem saeculum integrum huic examini, quocumque modo hactenus cognito, sufficiat.” Rose Ball concludes from this, “From the last clause it would appear that he did not know how the result was demonstrated.”

  13. Clearly, this text refers to Paul Tannery, who at about this time was preparing editions of the works of Fermat and Descartes, not to his brother Jules Tannery.

  14. “… valeurs qu’il tenait, supposent certains, de Fermat lui-même” [274].

  15. Derrick Henry Lehmer (1905–1991), husband of Emma Markovna Lehmer (1906–2007), who should not be confused with his father Derrick Norton Lehmer (1867–1938), also a professor at the University of California at Berkeley, who also studied exactly the same kind of number theory. However, they are not systematically distinguished by the main databases: in MatSciNet, the works of D.N. Lehmer’s are not classified as such, but in ZBMath they are attributed to D.H. Lehmer. Therefore, the only way to differentiate them is to look at the texts of the articles themselves. The generalizations of perfect numbers mentioned below are D.N. Lehmer. However, in this case, we are talking about the works of D.H. Lehmer, which consituted the contents of his PhD in 1930.

  16. The village priest Ivan Pervushin was the eldest of 17 children in his family, which from childhood aroused his interest in prime numbers. However, Wikipedia claims that there were only 16 children in the family of his parents, which, of course, would explain his interest in powers of 2. Before the discovery of the Mersenne prime \({{M}_{{61}}}\) in 1877, he found a prime divisor of Fermat’s number \({{F}_{{12}}}\), and in 1878 the Fermat number \({{F}_{{23}}}\). Before him, only Euler and Clausen could find new divisors for Fermat numbers, and, simultaneously with him, Lucas.

  17. As the reviewer noted, in a situation where there are 17 children in a family, interest in prime numbers should arise in the junior child. The author missed this obvious consideration.

  18. I could not find Pervushin’s original publication, but only a mention of his findings in the Bulletin of the Petersburg Academy. A picturesque fragment from the report of Imshenetsky and Bunyakovsky is that “Tout en laissant à la charge de l’auteur la responsabilité pour l’exactitude du résultat qu’il a obtenu au bout de ses longs et fatigants calculs,—nous devons constater, pour sauvegarder son droit de priorité, que 1°. Le manuscrit du père Pervouchine contenant sa communication de l’année 1883, est déposé aux Archives de notre Académie; ce document est accompagné de quelques tables, calculés par l’auteur, et destinés a faciliter la vérification du résultat qu’il a obtenu” [178].

  19. “In addition, there was another article (Catalan) on this issue, but, unfortunately, I was deprived of the opportunity to get acquainted with this article, since I could not get this journal anywhere in Moscow” [11].

  20. We reproduce Powers’ obituary written by Lehmer for AMS, which mentions both of these circumstances: “This amateur mathematician died on Jan. 31, 1952, at Puente, California. He would have been 77 years old on April 27. Mr. Powers was more responsible than any other man for the demonstration of the failure of Mersenne’s conjecture. He proved that \({{2}^{{89}}} - 1\) and \({{2}^{{107}}} - 1\) were primes, and that several other Mersenne numbers were composite by long and laborious desk machine calculations. He was not aware of the discovery, the night before his death, of two new Mersenne primes (MTAC, vol. 6, p. 61). Mr. Powers was born in Fountain, Colorado, and spent most of his life in Denver.”

  21. The same Raphael Robinson, the famous logician and husband of Julia Robinson.

  22. The same Hans Riesel, who is known for his work on the factorization of numbers of the form \(k \cdot {{2}^{n}} \pm 1\): the Lucas–Lehmer–Riesel test, Riesel numbers, Riesel Sieve, etc.

  23. Noll has since continued to deal with factorizations; on his webpage (http://www.isthe.com/chongo/index.html) you can find interesting links on this matter.

  24. Imagine how much a month of running such a machine would cost at that time—I have hypotheses about how schoolchildren could get access to it, but I am embarrassed to utter them.

  25. See the official website https://www.mersenne.org/; the project name is pronounced gimps.

  26. I do not know if this was originally meant, but Prime95 has become a favorite tool for testing system stability: “Prime95 has been a popular choice for stress/torture testing a CPU since its introduction, especially with overclockers and system builders. Since the software makes heavy use of the processor’s integer and floating point instructions, it feeds the processor a consistent and verifiable workload to test the stability of the CPU and the L1/L2/L3 processor cache. Additionally, it uses all of the cores of a multi-CPU/multi-core system to ensure a high-load stress test environment.”

  27. Exaflops = quintillion = million million million floating point operations per second.

  28. A well-known historical anecdote is connected with this decomposition. At the meeting of the American Mathematical Society on October 31, 1903, Cole gave a lecture, during which he did not utter a single word, but simply multiplied 193 707 721 by 761 838 257 287 on the board. He later mentioned that it took him “three years of Sundays” to find these divisors.

  29. This is without even discussing the question which of the historical calculations were checked or repeated, how many errors there were, etc. “And that leaves five—Well, six actually. But the idea is the important thing!”

  30. Mathematica 11.3 on an HP EliteBook 830GS with an Intel Core i7-8550U 1.99-GHz processor.

  31. However, some believe that the first irreproachable primality proof of \({{M}_{{127}}}\) was given only in 1894 by Fauquembergue, but even in this case the record stood for 57 years! I do not make any judgments about this, but many authors still have great doubts about the calculations of Fauquembergue himself (see, for example, [167]).

  32. The only source known to me where the opinion is seriously expressed that the number of Mersenne primes is finite is an article by Golubev [131].

  33. In fact, this is a later reinterpretation. Catalan himself is much more careful: “Si l’on admet ces deux propositions, et si l’on observe que \({{2}^{2}} - 1\), \({{2}^{3}} - 1\), \({{2}^{7}} - 1\) sont aussi des nombres premiers, on a ce théoréme empirique: Jusqu’une certaine limite, si \({{2}^{n}} - 1\) est un nombre premier p, \({{2}^{p}} - 1\) est un nombre premier \(p'\), \({{2}^{{p'}}} - 1\) est un nombre premier \(p''\), etc. Cette proposition a quelque analogie avec le théoréme suivant, énoncé par Fermat, et dont Euler a montré l’inexactitude: Si n est une puissance de \(2\), \({{2}^{n}} + 1\) est un nombre premier.” Comparing this conjecture with Fermat’s conjecture that the Fermat numbers are prime, he directly hints that it may be wrong already at the next step (cf. [114, 132]).

  34. “Haec autem propter senarii numeri perfectionem eodem die sexiens repetito sex diebus perfecta narrantur, non quia Deo fuerit necessaria mora temporum, quasi qui non potuerit creare omnia simul, quae deinceps congruis motibus peragerent tempora; sed quia per senarium numerum est operam significata perfectio. Numerus quippe senarius primus completur suis partibus, id est sexta sui parte et tertia et dimidia, quae sunt unum et duo et tria, quae in summam ducta sex fiunt,” XI–XXX.

  35. Seeing these numbers in such a context, any specialist in exceptional numerology cannot help but start. After all, in fact, 56 = 2 · 28 is the dimension of the smallest representation of E7, and 248 = 496/2 is the dimension of the smallest representation of E8.

  36. To avoid doubt, I specify that this is Benjamin Peirce, 1809–1880, father of Charles Peirce, 1839–1914. In Russian, they usually shamelessly write “Pierce.”

  37. That Sylvester! In his old age, he suddenly began to experiment with the classical impenetrable problems of number theory, including Goldbach’s problem.

  38. The same Gradshtein, better known to Soviet mathematicians as Gradshtein–Ryzhik.

  39. More under some additional assumptions.

  40. I first heard about multiple perfect numbers from Nikolai Grigoryevich Chudakov in 1968. Then, at LOMI, and even in the Mathematics and Mechanics Faculty, letters from lovers of mathematics with new great discoveries came in a stream. They were written by hand on checkered pieces of paper torn from school notebooks. About 90% of those were proofs of Fermat’s theorem with the same standard error. But there were also more interesting things—a refutation of the Cantor diagonal process, a proof of the even Goldbach conjecture based on the equality 3 + 3 = 5, a proof of the formula (–1) ⋅ (–1) = –1, etc. Now, of course, all such delirium immediately spills into social networks, bypassing the Science Department of the Vasileostrovsky district committee of the CPSU (in fact, social networks now play the same role). So, Chudakov mentioned the letter, whose author found a general solution to the equation \(\sigma (n) = kn\) for any k and assured that this knowledge guarantees immortality in the literal physical sense: “Some pirates achieved immortality by great deeds of cruelty or derring-do … But the captain had long ago decided that he would, on the whole, prefer to achieve immortality by not dying.” Nikolai Grigorievich smiled and added: “This is not surprising, because already the one who finds all the solutions to the equation \(\sigma (n) = 2n\) will become immortal.” Much more about the role of Nikolai Grigorievich in the emergence of this article, as well as about how to avoid Thanatos and black Kera, is narrated in [379].

  41. The term “friendly pair” also exists, but it means something completely different, equality σ(m)/m = σ(n)/n.

  42. This view is widely accepted in the literature: “It might be argued that elementary number theory began with Pythagoras, who noted two-and-a-half millennia ago that 220 and 284 form an amicable pair.” [262].

  43. “Two hundred goats, twenty goats, two hundred sheep, twenty rams,” Genesis, XXXII, 14.

  44. Herman te Riele writes about this that “De meeste bekende bevriende getallenparen zijn gevonden met behulp van variaties van de Regel van Thabit ibn Kurrah” [674].

  45. Another Paganini, full namesake.

  46. However, Lee claims that Escott made mistakes and in fact discovered only 219 pairs.

  47. Here, however, one must carefully compare how they take into account repetitions, Iranian and Arabic authors, etc. But this, of course, is for a serious proper historical work.

  48. To illustrate what kind of nonsense the Internet is filled with, I will quote a tale about Fedor Ivanovich Duz-Khotimirsky, which was widely discussed on chess sites: he “… filled entire reams of paper with numbers, revealing ‘related numbers,’ … And mathematical geniuses sat in academies, and to one of them, an academician named Vinogradov, Uncle Fedya sent the ‘related numbers’ he had discovered in infinity. As I understand it, Descartes found the first 14 of these numbers in his time, and Uncle Fedya brought their number to 600. The academician, of course, was an intelligent person and published Duz’s discovery under his great name. Duz was terribly angry with him, but he did not sue and prove his authorship. Firstly, because he would have lost for sure. And secondly, because he did not want to appeal to the state, which he did not recognize in principle” [9]. Wow, six hundred pairs of amicable numbers by hand in a school notebook—this is a stronger joke than Goethe’s Micromegas.

  49. “Millionen stehen hinter mir.”

  50. We do not consider triples with repeating elements.

REFERENCES

  1. M. M. Artyukhov, “On problems of the theory of amicable numbers,” Acta Arith. 27, 281–291 (1975).

    MathSciNet  Google Scholar 

  2. W. Borho, “Befreundete Zahlen: Ein zweitausend Jahre altes Thema der elementaren Zahlentheorie,” in Lebendige Zahlen (Springer, Basel, 1985), pp. 5–38 [in German].

    MATH  Google Scholar 

  3. N. A. Vavilov, “Computers as a novel mathematical reality: I. A personal account,” Dokl. Math. 107 (2), 130–141 (2023).

    Article  MathSciNet  MATH  Google Scholar 

  4. N. A. Vavilov, “Computers as a novel mathematical reality: II. Waring’s problem,” Dokl. Math. 107 (3), (2023).

  5. N. A. Vavilov, “Computers as a novel mathematical reality: IV. Goldbach’s conjecture,” Dokl. Math. 107 (3), (2023).

  6. N. A. Vavilov and V. G. Khalin, Exercises for the Course “Mathematics and Computers,” Vol. 1: Arithmetics and Number Theory (OTsEiM, St. Petersburg, 2005) [in Russian].

  7. N. A. Vavilov and V. G. Khalin, Supplementary Exercises for the Course “Mathematics and Computers” (OTsEiM, St. Petersburg, 2007) [in Russian].

  8. N. A. Vavilov, V. G. Khalin, and A. V. Yurkov, Mathematica for Nonmathematicians (Mosk. Tsentr Neprer. Mat. Obrazovan., Moscow, 2021) [in Russian].

    Google Scholar 

  9. E. V. Vengerova, Memuareski (Tekst, Moscow, 2016) [in Russian].

    Google Scholar 

  10. S. Yates, Repunits and Repetends (Samuel Yates, 1982).

    MATH  Google Scholar 

  11. I. S. Gradstein, “On odd perfect numbers,” Mat. Sb. 32 (3), 476–510 (1925).

    Google Scholar 

  12. G. M. Edwards, Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory, 3rd. ed. (Springer, New York, 2000).

    MATH  Google Scholar 

  13. H. L. Abbott, C. E. Aull, E. Brown, and D. Suryanarayana, “Quasiperfect numbers,” Acta Arith. 22, 439–447 (1973). https://doi.org/10.4064/aa-22-4-439-447

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Alanen, O. Ore, and J. Stemple, “Systematic computations on amicable numbers,” Math. Comput. 21, 242–245 (1967). https://doi.org/10.1090/S0025-5718-1967-0222006-7

    Article  MathSciNet  MATH  Google Scholar 

  15. R. C. Archibald, “Mersenne’s numbers,” Scr. Math. 3, 112–119 (1935).

    MATH  Google Scholar 

  16. M. M. Artuhov, “On the problem of \(h\)-fold perfect numbers,” Acta. Arith. 23, 249–255 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Bach and J. Shallit, Algorithmic Number Theory: Efficient Algorithms (MIT Press, Cambridge, Mass., 1996), Vol. 1.

    MATH  Google Scholar 

  18. T. Bang, “Store primtal,” Nordisk Mat. Tidskr. 2, 157–168 (1954).

    MathSciNet  Google Scholar 

  19. C. B. Barker, “Proof that the Mersenne number \({{M}_{{167}}}\) is composite,” Bull. Am. Math. Soc. 51 (6), 389 (1945). https://doi.org/10.1090/S0002-9904-1945-08362-110.1090/S0002-9904-1945-08362-1

    Article  MATH  Google Scholar 

  20. P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff, Jr., “The new Mersenne conjecture,” Am. Math. Mon. 96 (2), 125–128 (1989). https://doi.org/10.1080/00029890.1989.11972155

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Battiato and W. Borho, “Are there odd amicable numbers not divisible by three?,” Math. Comput. 50, 633–637 (1988). https://doi.org/10.2307/2008630

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Battiato and W. Borho, “Breeding amicable numbers in abundance II,” Math. Comput. 70, 1329–1333 (2001). https://doi.org/10.1090/S0025-5718-00-01279-5

    Article  MathSciNet  MATH  Google Scholar 

  23. W. E. Beck and R. M. Najar, “Reduced and augmented amicable pairs to 108,” Fibonacci Q. 31, 295–298 (1993).

    MATH  Google Scholar 

  24. A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed. (Dover, New York, 1966).

  25. M. Benito and J. L. Varona, “Advances in aliquot sequences,” Math. Comput. 68 (225), 389–393 (1999). https://doi.org/10.1090/S0025-5718-99-00991-6

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Benito, W. Creyaufmüller, J. L. Varona, and P. Zimmermann, “Aliquot sequence 3630 ends after reaching 100 digits,” Exp. Math. 11 (2), 201–206 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  27. H. A. Bernhard, “On the least possible odd perfect number,” Am. Math. Mon. 56, 628–629 (1949). https://doi.org/10.2307/2304736

    Article  MathSciNet  MATH  Google Scholar 

  28. C. E. Bickmore, “On the numerical factors of \({{a}^{n}} - 1\),” Messenger Math. 25 (2), 1–44 (1895–1896).

    MATH  Google Scholar 

  29. C. E. Bickmore, “On the numerical factors of \({{a}^{n}} - 1\) (Second notice),” Messenger Math. 26, 1–38 (1896–1897).

    MATH  Google Scholar 

  30. K. Blankenagel and W. Borho, “New amicable four-cycles II,” Int. J. Math. Sci. Comput. 5 (1), 49–51 (2015).

    MathSciNet  MATH  Google Scholar 

  31. K. Blankenagel, W. Borho, and A. vom Stein, “New amicable four-cycles,” Math. Comput. 72 (244), 2071–2076 (2003). https://doi.org/10.1090/s0025-5718-03-01489-3

    Article  MathSciNet  MATH  Google Scholar 

  32. W. Borho, “Über die Fixpunkte der \(k\)-fach iterierten Teilersummenfunktion,” Mitt. Math. Gesellsch. Hamburg 9 (5), 34–48 (1969).

    MathSciNet  MATH  Google Scholar 

  33. W. Borho, “Bemerkung zu einer Arbeit von H.-J. Ka-nold,” J. Reine Angew. Math. 243, 219–220 (1970).

    MathSciNet  MATH  Google Scholar 

  34. W. Borho, “On Thabit ibn Kurrah’s formula for amicable numbers,” Math. Comput. 26, 571–578 (1972).

    MathSciNet  MATH  Google Scholar 

  35. W. Borho, “Befreundete Zahlen mit gegebener Primteileranzahl,” Math. Ann. 209, 183–193 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  36. W. Borho, “Eine Schranke für befreundete Zahlen mit gegebener Teileranzahl,” Math. Nachr. 63, 297–301 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  37. W. Borho, “Some large primes and amicable numbers,” Math. Comput. 36 (153), 303–304 (1981). https://doi.org/10.1090/S0025-5718-1981-0595068-2

    Article  MathSciNet  MATH  Google Scholar 

  38. W. Borho and K. Blankenagel, “Befreundete Zyklen,” Mitt. Math. Gesellsch. Hamburg 35, 67–75 (2015).

    MathSciNet  MATH  Google Scholar 

  39. W. Borho and H. Hoffmann, “Breeding amicable numbers in abundance,” Math. Comput. 46, 281–293 (1986). https://doi.org/10.1090/S0025-5718-1986-0815849-1

    Article  MathSciNet  MATH  Google Scholar 

  40. W. Bosma and B. Kane, “The aliquot constant,” Q. J. Math. 63 (2), 309–323 (2012). https://doi.org/10.1093/qmath/haq050

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Bratley, F. Lunnon, and J. McKay, “Amicable numbers and their distribution,” Math. Comput. 24, 431–432 (1970). https://doi.org/10.1090/S0025-5718-1970-0271005-8

    Article  MathSciNet  MATH  Google Scholar 

  42. P. Bratley and J. McKay, “More amicable numbers,” Math. Comput. 22, 677–678 (1968). https://doi.org/10.1090/S0025-5718-1968-0225706-9

    Article  MathSciNet  MATH  Google Scholar 

  43. A. Brauer, “On the non-existence of odd perfect numbers of form \({{p}^{\alpha }}q_{1}^{2}q_{2}^{2} \cdots q_{{t - 1}}^{2}q_{t}^{4}\),” Bull. Am. Math. Soc. 49, 712–718 (1943). https://doi.org/10.1090/S0002-9904-1943-08007-X10.1090/S0002-9904-1943-08007-X

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Brauer, “Note on the non-existence of odd perfect numbers of form \({{p}^{\alpha }}q_{1}^{2}q_{2}^{2} \cdots q_{{t - 1}}^{2}q_{t}^{4}\),” Bull. Am. Math. Soc. 49, 937 (1943). https://doi.org/10.1090/S0002-9904-1943-08067-610.1090/S0002-9904-1943-08067-6

    Article  MathSciNet  MATH  Google Scholar 

  45. R. P. Brent and G. L. Cohen, “A new lower bound for odd perfect numbers,” Math. Comput. 53, 431–437 (1989). https://doi.org/10.1090/S0025-5718-1989-0968150-2

    Article  MathSciNet  MATH  Google Scholar 

  46. R. P. Brent, G. L. Cohen, and H. J. J. te Riele, “Improved techniques for lower bounds for odd perfect numbers,” Math. Comput. 57 (196), 857–868 (1991). https://doi.org/10.1090/S0025-5718-1991-1094940-3

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Brentjes and J. P. Hogendijk, “Notes on Thābit ibn Qurra and his rule for amicable numbers,” Hist. Math. 16, 373–378 (1989). https://doi.org/10.1016/0315-0860(89)90084-0

    Article  MATH  Google Scholar 

  48. J. Brillhart, “Some miscellaneous factorizations,” Math. Comput. 17, 447–450 (1963).

    Article  MATH  Google Scholar 

  49. J. Brillhart, “On the factors of certain Mersenne numbers. II,” Math. Comput. 18, 87–92 (1964). https://doi.org/10.1090/S0025-5718-1964-0159776-X

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Brillhart and G. D. Johnson, “On the factors of certain Mersenne numbers,” Math. Comput. 14, 365–369 (1960). https://doi.org/10.1090/S0025-5718-1960-0123507-6

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Brillhart, D. H. Lehmer, and J. L. Selfridge, “New primality criteria and factorizations of \({{2}^{m}} \pm 1\),” Math. Comput. 29, 620–647 (1975).

    MathSciNet  MATH  Google Scholar 

  52. J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of \({{b}^{n}} \pm 1\), \(b = 2,3,5,6,7,10,11,12\) up to High Powers, 2nd ed. (Am. Math. Soc., Providence, RI, 1988).

    MATH  Google Scholar 

  53. J. Brillhart and J. L. Selfridge, “Some factorizations of \({{2}^{n}} \pm 1\) and related results,” Math. Comput. 21, 87–96 (1967). Corrigendum, Math. Comput. 21, 751 (1967).

    Article  MATH  Google Scholar 

  54. J. W. Bruce, “A really trivial proof of the Lucas–Lehmer test,” Am. Math. Mon. 100 (4), 370–371 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  55. R. D. Carmichael, “Note on multiply perfect numbers,” Am. Math. Soc. Bull. 17 (2), 518 (1911).

    MATH  Google Scholar 

  56. E. Catalan, “Propositions et questions diverses,” Bull. Soc. Math. France 16, 128–129 (1888).

    Article  MathSciNet  Google Scholar 

  57. P. A. Cataldi, Trattato dè numeri perfetti (Giovanni Rossi, Bologna, 1603) [in Italian].

    Google Scholar 

  58. P. Cattaneo, “Sui numeri quasiperfetti,” Boll. Un. Mat. Ital. 6, 59–62 (1951).

    MATH  Google Scholar 

  59. C. Shi-Chao and L. Hao, “Bounds for odd \(k\)-perfect numbers,” Bull. Aust. Math. Soc. 84 (3), 475–480 (2011). https://doi.org/10.1017/S000497271100246210.1017/S0004972711002462

    Article  MathSciNet  MATH  Google Scholar 

  60. C. Shi-Chao and L. Hao, “Odd multiperfect numbers,” Bull. Aust. Math. Soc. 88 (1), 56–63 (2013). https://doi.org/10.1017/S0004972712000858

    Article  MathSciNet  MATH  Google Scholar 

  61. C. Yong-Gao and T. Cui-E, “Improved upper bounds for odd multiperfect numbers,” Bull. Aust. Math. Soc. 89 (3), 353–359 (2014). https://doi.org/10.1017/S0004972713000488

    Article  MathSciNet  MATH  Google Scholar 

  62. Y. Chishiki, T. Goto, and Y. Ohno, “On the largest prime divisor of an odd harmonic number,” Math. Comput. 76 (259), 1577–1587 (2007). https://doi.org/10.1090/S0025-5718-07-01933-3

    Article  MathSciNet  MATH  Google Scholar 

  63. K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, “Numerical and statistical analysis of aliquot sequences,” Exp. Math. 29 (4), 414–425 (2020). https://doi.org/10.1080/10586458.2018.1477077

    Article  MathSciNet  MATH  Google Scholar 

  64. G. L. Cohen, “On odd perfect numbers: II. Multiperfect numbers and quasiperfect numbers,” J. Aust. Math. Soc. Ser. A 29, 369–384 (1980). https://doi.org/10.1017/S1446788700021376

    Article  MathSciNet  MATH  Google Scholar 

  65. G. L. Cohen, “Even perfect numbers,” Math. Gaz. 65, 28–30 (1981). https://doi.org/10.2307/3617930

    Article  MathSciNet  MATH  Google Scholar 

  66. G. L. Cohen, “The non-existence of quasiperfect numbers of certain forms,” Fibonacci Q. 20, 81–84 (1982).

    MATH  Google Scholar 

  67. G. L. Cohen, “On primitive abundant numbers,” J. Aust. Math. Soc. Ser. A 34, 123–137 (1983). https://doi.org/10.1017/S1446788700019819

    Article  MathSciNet  MATH  Google Scholar 

  68. G. L. Cohen, “On the largest component of an odd perfect number,” J. Aus. Math. Soc. Ser. A 42, 280–286 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  69. G. L. Cohen, S. F. Gretton, and P. Hagis, Jr., “Multiamicable numbers,” Math. Comput. 64 (212), 1743–1753 (1995). https://doi.org/10.1017/S1446788700028251

    Article  MathSciNet  MATH  Google Scholar 

  70. G. L. Cohen and P. Hagis, Jr., “Results concerning odd multiperfect numbers,” Bull. Malays. Math. Soc. 8 (1), 23–26 (1985).

    MathSciNet  MATH  Google Scholar 

  71. G. L. Cohen and M. D. Hendy, “On odd multiperfect numbers,” Math. Chron. 10, 57–61 (1981).

    MATH  Google Scholar 

  72. G. L. Cohen and H. J. J. te Riele, “Iterating the sum-of-divisors function,” Exp. Math. 5 (2), 91–100 (1996). Errata: Exp. Math. 6 (2), 177 (1997).

    Google Scholar 

  73. G. L. Cohen and H. J. J. te Riele, “On \(\phi \)-amicable pairs,” Math. Comput. 67 (221), 399–411 (1998).

    Article  MATH  Google Scholar 

  74. G. L. Cohen and R. M. Sorli, “On the number of distinct prime factors of an odd perfect number: Combinatorial algorithms,” J. Discrete Algorithms 1 (1), 21–35 (2003). https://doi.org/10.1016/S1570-8667(03)00004-2

    Article  MathSciNet  MATH  Google Scholar 

  75. G. L. Cohen and R. M. Sorli, “Odd harmonic numbers exceed 1024,” Math. Comput. 79 (272), 2451–2460 (2010). https://doi.org/10.1090/S0025-5718-10-02337-9

    Article  MATH  Google Scholar 

  76. G. L. Cohen and R. M. Sorli, “On odd perfect numbers and even 3-perfect numbers,” Integers 12 (6), 1213–1230 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  77. G. L. Cohen and R. J. Williams, “Extensions of some results concerning odd perfect numbers,” Fibonacci Q. 23, 70–76 (1985).

    MathSciNet  MATH  Google Scholar 

  78. H. Cohen, “On amicable and sociable numbers,” Math. Comput. 24, 423–429 (1970). https://doi.org/10.1090/S0025-5718-1970-0271004-6

    Article  MathSciNet  MATH  Google Scholar 

  79. H. Cohen, A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1993).

    Book  MATH  Google Scholar 

  80. P. Cohen, K. Cordwell, A. Epstein, K. Chung-Hang, A. Lott, and S. J. Miller, “On near-perfect numbers,” Acta Arith. 194 (4), 341–366 (2020). https://doi.org/10.4064/aa180821-11-10

    Article  MathSciNet  MATH  Google Scholar 

  81. F. N. Cole, “On the factoring of large numbers,” Bull. Am. Math. Soc. 10, 134–137 (1903).

    Article  MathSciNet  Google Scholar 

  82. W. N. Colquitt and L. Welsh, Jr., “A new Mersenne prime,” Math. Comput. 56 (194), 867–870 (1991). https://doi.org/10.1090/S0025-5718-1991-1068823-9

    Article  MathSciNet  MATH  Google Scholar 

  83. J. H. Conway and R. K. Guy, The Book of Numbers (Copernicus, New York, 1996).

    Book  MATH  Google Scholar 

  84. R. J. Cook, “Bounds for odd perfect numbers,” in Number Theory (Am. Math. Soc., Providence, RI, 1999), pp. 67–71.

    Google Scholar 

  85. P. Costello, “Amicable pairs of Euler’s first form,” J. Rec. Math. 10, 183–189 (1977–1978).

    MATH  Google Scholar 

  86. P. Costello, “Amicable pairs of the form (i, 1),” Math. Comput. 56, 859–865 (1991). https://doi.org/10.1090/S0025-5718-1991-1068822-7

    Article  MathSciNet  MATH  Google Scholar 

  87. P. Costello and R. A. C. Edmonds, “Gaussian amicable pairs,” Missouri J. Math. Sci. 30 (2), 107–116 (2018). https://doi.org/10.35834/mjms/1544151688

    Article  MathSciNet  MATH  Google Scholar 

  88. R. Crandall and M. A. Penk, “A search for large twin prime pairs,” Math. Comput. 33, 383–388 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  89. R. Crandall and C. Pommerance, Prime Numbers: A Computational Perspective, 2nd ed. (Springer, New York, 2005).

    Google Scholar 

  90. J. T. Cross, “A note on almost perfect numbers,” Math. Mag. 47, 230–231 (1974). https://doi.org/10.2307/2689220

    Article  MathSciNet  MATH  Google Scholar 

  91. A. Cunningham, “On Mersenne’s numbers,” Proc. London Math. Soc. s2-9 (1) (1911).

  92. A. Cunningham, “On Mersenne’s numbers,” in Proceedings of the 5th International Congress of Mathematicians (Cambridge Univ. Press, Cambridge, 1913), Vol. 1, pp. 384–386; Brit. Ass. Rep. Dundee 82, 406–407 (1913).

  93. A. Cunningham, “On Lucas’s process applied to composite Mersenne’s numbers,” London Math. Soc. Proc. 32 (2), 17 (1919).

    MATH  Google Scholar 

  94. L. Dai, H. Pan, and C. Tang, “Note on odd multiperfect numbers,” Bull. Aust. Math. Soc. 87 (3), 448–451 (2013). https://doi.org/10.1017/S0004972712000731

    Article  MathSciNet  MATH  Google Scholar 

  95. G. G. Dandapat, J. L. Hunsucker, and C. Pomerance, “Some new results on odd perfect numbers,” Pac. J. Math. 57, 359–364 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  96. J. S. Devitt, R. K. Guy, and J. L. Selfridge, “Third report on aliquot sequences,” Proceedings of the 6th Manitoba Conference on Numerical Mathematics, Univ. Manitoba, Winnipeg, Man., 1976 (Congress. Numer., Winnipeg, Man., 1977), pp. 177–204.

  97. K. Devlin, Mathematics: The New Golden Age, 2nd ed. (Columbia Univ. Press, New York, 1999).

    MATH  Google Scholar 

  98. L. E. Dickson, “Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors,” Am. J. Math. 35, 413–422 (1913). https://doi.org/10.2307/2370405

    Article  MATH  Google Scholar 

  99. L. E. Dickson, “Theorems and tables on the sum of the divisors of a number,” Q. J. Math. 44, 264–296 (1913).

    MATH  Google Scholar 

  100. L. E. Dickson, “Amicable number triples,” Am. Math. Mon. 20, 84–91 (1913). https://doi.org/10.1080/00029890.1913.11997926

    Article  MathSciNet  MATH  Google Scholar 

  101. L. E. Dickson, History of the Theory of Numbers (Chelsea, New York, 1952), Vol. 1.

    Google Scholar 

  102. J. Dieudonné, Pour l’honneur de l’esprit humain. Les mathématiques aujourd’hui. Histoire et Philosophie des Sciences (Librairie Hachette, Paris, 1987) [in French].

    MATH  Google Scholar 

  103. S. Drake, “The rule behind ‘Mersenne’s numbers’,” Phys. Riv. Int. Stor. Sci. 13, 421–424 (1971).

    MathSciNet  Google Scholar 

  104. H. Dubner, “New amicable pair record,” October 14, 1997. https://listserv.nodak.edu/scripts/wa.exe?A2= ind9710&L=NMBRTHRY&F=&S=&Pi5. Accessed November 14, 2020.

  105. J. R. Ehrman, “The number of prime divisors of certain Mersenne numbers,” Math. Comput. 21, 700–704 (1967). https://doi.org/10.1090/S0025-5718-1967-0223320-1

    Article  MathSciNet  Google Scholar 

  106. Z. Engberg and P. Pollack, “The reciprocal sum of divisors of Mersenne numbers,” Acta Arith. 197 (4), 421–440 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  107. P. Erdős, “On amicable numbers,” Publ. Math. Debrecen 4, 108–111 (1955–1956).

    Article  MathSciNet  MATH  Google Scholar 

  108. P. Erdős, “On perfect and multiply perfect numbers,” Ann. Mat. Pura Appl. 42 (4), 253–258 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  109. P. Erdős, “On the sum \(\sum\nolimits_{d{{{|2}}^{n}} - 1} {{d}^{{ - 1}}}\),” Isr. J. Math. 9, 43–48 (1971).

    Google Scholar 

  110. P. Erdős, “On asymptotic properties of aliquot sequences,” Math. Comput. 30, 641–645 (1976). https://doi.org/10.1090/S0025-5718-1976-0404115-8

    Article  MathSciNet  MATH  Google Scholar 

  111. P. Erdős, P. Kiss, and C. Pomerance, “On prime divisors of Mersenne numbers,” Acta Arith. 57, 267–281 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  112. E. B. Escott, “Amicable numbers,” Scr. Math. 12, 61–72 (1946).

    MathSciNet  MATH  Google Scholar 

  113. L. Euler, “De numeris amicabilibus,” in Opuscula varii argumenti (1750), pp. 23–107.

  114. I. S. Eum, “A congruence relation of the Catalan–Mersenne numbers,” Indian J. Pure Appl. Math. 49 (3), 521–526 (2018). https://doi.org/10.1007/s13226-018-0281-8

    Article  MathSciNet  Google Scholar 

  115. E. Fauquembergue, “Nombres de Mersenne,” Sphinx-Œdipe 9, 103–105 (1914); 15, 17–18 (1920).

  116. E. Fauquembergue, “Plus grand nombre premier connu (question 176, de G. de Rocquigny),” Interméd. Math. 24, 33 (1917).

    MATH  Google Scholar 

  117. A. Ferrier, Table donnant, jusqu'à 100.000, les nombres premiers et les nombres composés n’ayant pas de diviseur inférieur à 17, avec, pour chacun d’eux, son plus petit diviseur (Librairie Vuibert, Paris, 1947) [in French].

    Google Scholar 

  118. A. Flammenkamp, “New sociable number,” Math. Comput. 56, 871–873 (1991). https://doi.org/10.1090/S0025-5718-1991-1052094-3

    Article  MathSciNet  MATH  Google Scholar 

  119. S. Fletcher, P. P. Nielsen, and P. Ochem, “Sieve methods for odd perfect numbers,” Math. Comput. 81 (279), 1753–1776 (2012). https://doi.org/10.1090/S0025-5718-2011-02576-7

    Article  MathSciNet  MATH  Google Scholar 

  120. B. Franqui and M. García, “Some new multiply perfect numbers,” Am. Math. Mon. 60, 459–462 (1953). https://doi.org/10.1080/00029890.1953.11988324

    Article  MathSciNet  MATH  Google Scholar 

  121. B. Franqui and M. García, “57 new multiply perfect numbers,” Scr. Math. 20, 169–171 (1954).

    MathSciNet  MATH  Google Scholar 

  122. M. García, “New amicable pairs,” Scr. Math. 23, 167–171 (1957).

    MathSciNet  MATH  Google Scholar 

  123. M. García, “New amicable pairs of Euler’s first form with greatest common factor a prime times a power of 2,” Nieuw Arch. Wisk. 4 17 (1), 25–27 (1999).

  124. M. García, “A million new amicable pairs,” J. Integer Seq. 4 (2), 01.2.6 (2001).

  125. V. García, “The first known type (7, 1) amicable pair,” Math. Comput. 72 (242), 939–940 (2003). https://doi.org/10.1090/S0025-5718-02-01450-3

    Article  MathSciNet  MATH  Google Scholar 

  126. M. García, J. M. Pedersen, and H. J. J. te Riele, “Amicable pairs, a survey,” Fields Inst. Commun. 41, 179–196 (2004).

    MathSciNet  MATH  Google Scholar 

  127. M. Gardner, “Perfect, amicable, sociable,” in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American (Vintage, New York, 1978), pp. 160–171.

    Google Scholar 

  128. D. B. Gillies, “Three new Mersenne primes and a statistical theory,” Math. Comput. 18 (85), 93–97 (1964). https://doi.org/10.1090/S0025-5718-1964-0159774-6

    Article  MathSciNet  MATH  Google Scholar 

  129. S. Gimbel and J. H. Jaroma, “Sylvester: Ushering in the modern era of research on odd perfect numbers,” Integers: Electron. J. Combinatorial Number Theory 3, 1–26 (2003).

    MathSciNet  MATH  Google Scholar 

  130. A. A. Gioia and A. M. Vaidya, “Amicable numbers with opposite parity,” Am. Math. Mon. 74, 969–973 (1967). https://doi.org/10.2307/2315280

    Article  MathSciNet  MATH  Google Scholar 

  131. V. A. Golubev, “Nombres de Mersenne et caractères du nombre \(2\),” Mathesis 67, 257–262 (1958).

    MathSciNet  MATH  Google Scholar 

  132. I. J. Good, “Conjectures concerning the Mersenne numbers,” Math. Comput. 9, 120–121 (1955). https://doi.org/10.1090/S0025-5718-1955-0071444-6

    Article  MathSciNet  MATH  Google Scholar 

  133. T. Goto and Y. Ohno, “Odd perfect numbers have a prime factor exceeding 108,” Math. Comput. 77 (263), 1859–1868 (2008). https://doi.org/10.1090/S0025-5718-08-02050-4

    Article  MATH  Google Scholar 

  134. A. Granville, Number Theory Revealed: A Masterclass (Am. Math. Soc., Providence, RI, 2019).

    MATH  Google Scholar 

  135. O. Grün, “Über ungerade vollkommene Zahlen,” Math. Z. 55, 353–354 (1952). https://doi.org/10.1007/BF01181133

    Article  MathSciNet  MATH  Google Scholar 

  136. A. W. P. Guy and R. K. Guy, “A record aliquot sequence,” in Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, 50th Anniversary Symposium, Vancouver, BC, 1993 (Am. Math. Soc., Providence, RI, 1994), pp. 557–559.

  137. R. K. Guy, “The strong law of small numbers,” Am. Math. Mon. 95 (8), 697–712 (1988). https://doi.org/10.1080/00029890.1988.11972074

    Article  MathSciNet  MATH  Google Scholar 

  138. R. K. Guy, “Graphs and the strong law of small numbers,” in Graph Theory, Combinatorics, and Applications (Kalamazoo, MI, 1988) (Wiley, New York, 1991), Vol. 2, pp. 597–614.

    Google Scholar 

  139. R. K. Guy, “The second strong law of small numbers,” Math. Mag. 63 (1), 3–20 (1990). https://doi.org/10.1080/0025570X.1990.11977475

    Article  MathSciNet  MATH  Google Scholar 

  140. R. K. Guy, “Unsolved problems in number theory,” in Problem Books in Mathematics, 3rd ed. (Springer, New York, 2004).

    Google Scholar 

  141. R. K. Guy, D. H. Lehmer, J. L. Selfridge, and M. C. Wunderlich, “Second report on aliquot sequences,” in Proceedings of the 3rd Manitoba Conference on Numerical Mathematics (Utilitas Math., Winnipeg, Man., 1974), pp. 357–368.

  142. R. K. Guy and J. L. Selfridge, “Interim report on aliquot series,” in Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. of Manitoba, Winnipeg, Man., 1971), pp. 557–580.

  143. R. K. Guy and J. L. Selfridge, “What drives an aliquot sequence?,” Math. Comput. 29, 101–107 (1975). https://doi.org/10.1090/S0025-5718-1975-0384669-X

    Article  MathSciNet  MATH  Google Scholar 

  144. P. Hagis, Jr., “On relatively prime odd amicable numbers,” Math. Comput. 23, 539–543 (1969). https://doi.org/10.1090/S0025-5718-1969-0246816-7

    Article  MathSciNet  MATH  Google Scholar 

  145. P. Hagis, Jr., “Lower bounds for relatively prime amicable numbers of opposite parity,” Math. Comput. 24, 963–968 (1970). https://doi.org/10.1090/S0025-5718-1973-0325507-9

    Article  MathSciNet  MATH  Google Scholar 

  146. P. Hagis, Jr., “Unitary amicable numbers,” Math. Comput. 25, 915–918 (1971). https://doi.org/10.1090/S0025-5718-1971-0299551-2

    Article  MathSciNet  MATH  Google Scholar 

  147. P. Hagis, Jr., “Relatively prime amicable numbers with twenty-one prime divisors,” Math. Mag. 45, 21–26 (1972). https://doi.org/10.1090/S0025-5718-1970-0276167-4

    Article  MathSciNet  MATH  Google Scholar 

  148. P. Hagis, Jr., “A lower bound for the set of odd perfect numbers,” Math. Comput. 35, 1027–1031 (1973). https://doi.org/10.1090/S0025-5718-1973-0325507-9

    Article  MathSciNet  MATH  Google Scholar 

  149. P. Hagis, Jr., “On the number of prime factors of a pair of relatively prime amicable numbers,” Math. Mag. 48, 263–266 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  150. P. Hagis, Jr., “On the largest prime divisor of an odd perfect number,” Math. Comput. 29, 922–924 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  151. P. Hagis, Jr., “Outline of a proof that every odd perfect number has at least eight prime factors,” Math. Comput. 35, 1027–1031 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  152. P. Hagis, Jr., “On the second largest prime divisor of an odd perfect number,” in Analytic Number Theory (Springer-Verlag, Berlin, 1981), pp. 254–263. https://doi.org/10.1007/BFb0096466

    Book  MATH  Google Scholar 

  153. P. Hagis, Jr., “Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors,” Math. Comput. 40, 399–404 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  154. P. Hagis, Jr., “Lower bounds for unitary multiperfect numbers,” Fibonacci Q. 22 (2), 140–143 (1984). https://doi.org/10.1090/S0025-5718-1970-0276167-4

    Article  MathSciNet  MATH  Google Scholar 

  155. P. Hagis, Jr., “The third largest prime factor of an odd multiperfect number exceeds 100,” Bull. Malays. Math. Soc. 9 (2), 43–49 (1986).

    MathSciNet  MATH  Google Scholar 

  156. P. Hagis, Jr., “A systematic search for unitary hyperperfect numbers,” Fibonacci Q. 25 (1), 6–10 (1987).

    MathSciNet  MATH  Google Scholar 

  157. P. Hagis, Jr., “Odd nonunitary perfect numbers,” Fibonacci Q. 28 (1), 11–15 (1990).

    MathSciNet  MATH  Google Scholar 

  158. P. Hagis, Jr., “A new proof that every odd triperfect number has at least twelve prime factors. A tribute to Emil Grosswald: Number theory and related analysis,” Contemp. Math. 143, 445–450 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  159. P. Hagis, Jr., and G. L. Cohen, “Some results concerning quasiperfect numbers,” J. Aust. Math. Soc., Ser. A 33, 275–286 (1982). https://doi.org/10.1017/S1446788700018401

    Article  MATH  Google Scholar 

  160. P. Hagis, Jr., and G. L. Cohen, “Every odd perfect number has a prime factor which exceeds 106,” Math. Comput. 67 (223), 1323–1330 (1998).

    Article  MATH  Google Scholar 

  161. P. Hagis, Jr., and G. Lord, “Quasi-amicable numbers,” Math. Comput. 31 (138), 608–611 (1977). https://doi.org/10.1090/S0025-5718-1977-0434939-3

    Article  MathSciNet  MATH  Google Scholar 

  162. P. Hagis, Jr., and W. L. McDaniel, “A new result concerning the structure of odd perfect numbers,” Proc. Am. Math. Soc. 32, 13–15 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  163. P. Hagis, Jr., and W. L. McDaniel, “On the largest prime divisor of an odd perfect number,” Math. Comput. 27, 955–957 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  164. P. Hagis, Jr., and W. L. McDaniel, “On the largest prime divisor of an odd perfect number II,” Math. Comput. 29, 922–924 (1975). https://doi.org/10.1090/S0025-5718-1975-0371804-2

    Article  MathSciNet  MATH  Google Scholar 

  165. H. Harborth, “Eine Bemerkung zu den vollkommenen Zahlen,” Elem. Math. 31, 115–117 (1976).

    MathSciNet  MATH  Google Scholar 

  166. K. G. Hare, “More on the total number of prime factors of an odd perfect number,” Math. Comput. 74, 1003–1008 (2005). https://doi.org/10.1090/S0025-5718-04-01683-7

    Article  MathSciNet  MATH  Google Scholar 

  167. G. Haworth, Mersenne Numbers (Univ. of Reading, Reading, UK, 1990).

    Google Scholar 

  168. D. R. Brown, “Odd perfect numbers,” Math. Proc. Cambridge Philos. Soc. 115, 191–196 (1994). https://doi.org/10.1017/S0305004100072030

    Article  MathSciNet  MATH  Google Scholar 

  169. M. R. Heyworth, “A conjecture on Mersenne’s conjecture,” N. Z. Math. Mag. 19, 147–151 (1982).

    MathSciNet  MATH  Google Scholar 

  170. M. R. Heyworth, “Continued fractions in a search for odd perfect numbers,” N. Z. Math. Mag. 19, 63–69 (1982).

    MathSciNet  MATH  Google Scholar 

  171. J. P. Hogendijk, “Thābit ibn Qurra and the pair of amicable numbers 17296; 18416,” Hist. Math. 12, 269–273 (1985). https://doi.org/10.1016/0315-0860(85)90025-4

    Article  MathSciNet  MATH  Google Scholar 

  172. J. A. Holdener, “A theorem of Touchard and the form of odd perfect numbers,” Am. Math. Mon. 109, 661–663 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  173. B. Hornfeck and E. Wirsing, “Über die Häufigkeit vollkommener,” Zahlen Math. Ann. 133, 431–438 (1957).

    Article  MATH  Google Scholar 

  174. A. Hurwitz, “New Mersenne primes,” Math. Comput. 16, 249–251 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  175. D. E. Iannucci, “The second largest prime divisor of an odd perfect number exceeds ten thousand,” Math. Comput. 68, 1749–1760 (1999). https://doi.org/10.1090/S0025-5718-99-01126-6

    Article  MathSciNet  MATH  Google Scholar 

  176. D. E. Iannucci, “The third largest prime divisor of an odd perfect number exceeds one hundred,” Math. Comput. 69, 867–879 (2000). https://doi.org/10.1090/S0025-5718-99-01127-8

    Article  MathSciNet  MATH  Google Scholar 

  177. D. E. Iannucci and R. M. Sorli, “On the total number of prime factors of an odd perfect number,” Math. Comput. 72 (244), 2077–2084 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  178. V. Imchenetski and V. Bouniakowsky, “Sur un nouveau nombre premier, annoncé par le père Pervouchine,” Extrait d’un rapport à l’Académie Bull. Acad. Impériale Sci. 31, 532–533 (1887). https://archive.org/stream/mobot31753003685184#page/n3 /mode /1up/search/Den. Accessed November 14, 2020.

  179. P. M. Jenkins, “Odd perfect numbers have a prime factor exceeding 107,” Math. Comput. 72, 1549–1554 (2003).

    Article  MATH  Google Scholar 

  180. R. P. Jerrard and N. Temperley, “Almost perfect numbers,” Math. Mag. 46, 84–87 (1973). https://doi.org/10.1080/0025570X.1973.11976282

    Article  MathSciNet  MATH  Google Scholar 

  181. P. Jobling, “Large amicable pairs,” June 6, 2004. https://listserv.nodak.edu/scripts/wa.exe?A2= ind0306&L=nmbrthry&P=R97&D=0. Accessed November 14, 2020.

  182. P. A Jobling, “New largest known amicable pair,” March 10, 2005. https://listserv.nodak.edu/cgi-bin/wa.exe?A2= ind0503&L=nmbrthry&F=&S=&PU2. Accessed November 14, 2020.

  183. H.-J. Kanold, “Untersuchungen über ungerade vollkommene Zahlen,” J. Reine Angew. Math. 183, 98–109 (1941).

    Article  MathSciNet  MATH  Google Scholar 

  184. H.-J. Kanold, “Folgerungen aus dem vorkommen einer Gaußschen Primzahl in der Primfaktorzerlegung einer ungeraden vollkommenen Zahl,” J. Reine Angew. Math. 186, 25–29 (1949).

    Article  MATH  Google Scholar 

  185. H.-J. Kanold, “Über befreundete Zahlen I,” Math. Nachr. 9, 243–248 (1953). https://doi.org/10.1002/mana.19530090408

    Article  MathSciNet  MATH  Google Scholar 

  186. H.-J. Kanold, “Über befreundete Zahlen II,” Math. Nachr. 10, 99–111 (1953). https://doi.org/10.1002/mana.19530100108

    Article  MathSciNet  MATH  Google Scholar 

  187. H.-J. Kanold, “Über mehrfach vollkommene Zahlen,” J. Reine Angew. Math. 197, 82–96 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  188. H.-J. Kanold, “Über ‘quasi-vollkommene Zahlen’,” Abh. Braunschweig Wiss. Ges. 40, 17–20 (1988).

    MathSciNet  MATH  Google Scholar 

  189. I. Kaplansky, “Lucas’ tests for Mersenne numbers,” Am. Math. Mon. 52, 188–190 (1945).

    Article  MathSciNet  MATH  Google Scholar 

  190. E. Karst, “New factors of Mersenne numbers,” Math. Comput. 15, 51 (1961). https://doi.org/10.1090/S0025-5718-1961-0116481-0

    Article  MathSciNet  MATH  Google Scholar 

  191. E. Karst, “Faktorenzerlegung Mersennescher Zahlen mittels programmgesteuerter Rechengeräte,” Numer. Math. 3, 79–86 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  192. E. Karst, “Search limits on divisors of Mersenne numbers,” Nordisk Tidskr. Informationsbehandling 2, 224–227 (1962).

    MathSciNet  MATH  Google Scholar 

  193. E. Karst, “A remarkable quartic yielding certain divisors of Mersenne numbers,” Nordisk Tidskr. Informationsbehandling 3, 122–123 (1963).

    MathSciNet  MATH  Google Scholar 

  194. M. Kishore, “On odd perfect, quasiperfect, and odd almost perfect numbers,” Math. Comput. 36, 583–586 (1981). https://doi.org/10.2307/2007662

    Article  MathSciNet  MATH  Google Scholar 

  195. M. Kishore, “Odd perfect numbers not divisible by 3,” Math. Comput. 40, 405–411 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  196. M. Kobayashi, P. Pollack, and C. Pomerance, “On the distribution of sociable numbers,” J. Number Theory 129 (8), 1990–2009 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  197. P. Acquaah and S. Konyagin, “On prime factors of odd perfect numbers,” Int. J. Number Theory 8 (6), 1537–1540 (2012). https://doi.org/10.1142/S1793042112500935

    Article  MathSciNet  MATH  Google Scholar 

  198. M. Kozek, F. Luca, P. Pollack, and C. Pomerance, “Harmonious pairs,” Int. J. Number Theory 11 (5), 1633–1651 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  199. M. Kraitchik, “Factorisation de \({{2}^{n}} \pm 1\),” Sphinx 8, 148–150 (1938).

    MATH  Google Scholar 

  200. M. Kraitchik, “On the factorization of \({{2}^{n}} \pm 1\),” Scr. Math. 18, 39–52 (1952).

    MathSciNet  MATH  Google Scholar 

  201. S. Kravitz, “Divisors of Mersenne numbers \(10,000 < p < 15,000\),” Math. Comput. 15, 292–293 (1961).

    MathSciNet  MATH  Google Scholar 

  202. U. Kühnel, “Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen,” Math. Z. 52, 202–211 (1949). https://doi.org/10.1007/BF02230691

    Article  MathSciNet  MATH  Google Scholar 

  203. E. J. Lee, “Amicable numbers and the bilinear Diophantine equation,” Math. Comput. 22, 181–187 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  204. E. J. Lee, “On divisibility by nine of the sums of even amicable pairs,” Math. Comput. 23, 545–548 (1969). https://doi.org/10.1090/S0025-5718-1969-0248074-6

    Article  MathSciNet  MATH  Google Scholar 

  205. E. J. Lee and J. S. Madachy, “The history and discovery of amicable numbers 1,” J. Recreational Math. 5 (2), 77–93 (1972); errata: 6 (2), 164 (1973); errata: 6 (3), 229 (1973).

  206. E. J. Lee and J. S. Madachy, “The history and discovery of amicable numbers 2,” J. Recreational Math. 5 (3), 153–173 (1972).

    MathSciNet  MATH  Google Scholar 

  207. E. J. Lee and J. S. Madachy, “The history and discovery of amicable numbers 3,” J. Recreational Math. 5 (4), 231–249 (1972).

    MathSciNet  MATH  Google Scholar 

  208. D. H. Lehmer, “Note on the Mersenne number \({{2}^{{139}}} - 1\),” Bull. Am. Math. Soc. 32, 522 (1926).

    Article  MathSciNet  MATH  Google Scholar 

  209. D. H. Lehmer, “Note on the largest Mersenne number,” Bull. Am. Math. Soc. 33, 271 (1927).

    MATH  Google Scholar 

  210. D. H. Lehmer, “An extended theory of Lucas’ functions,” Ann. Math. 31, 419–448 (1930).

    Article  MathSciNet  MATH  Google Scholar 

  211. D. H. Lehmer, “Note on Mersenne numbers,” Bull. Am. Math. Soc. 38, 383–384 (1932). https://doi.org/10.1090/S0002-9904-1932-05396-4

    Article  MathSciNet  MATH  Google Scholar 

  212. D. H. Lehmer, “Some new factorizations of \({{2}^{n}} \pm 1\),” Bull. Am. Math. Soc. 39, 105–108 (1933).

    Article  MathSciNet  MATH  Google Scholar 

  213. D. H. Lehmer, “Hunting big game in the theory of numbers,” Scr. Math. 1, 229–235 (1932–1933).

    MATH  Google Scholar 

  214. D. H. Lehmer, “On Lucas’ test for the primality of Mersenne numbers,” J. London Math. Soc. 10, 162–165 (1935). https://doi.org/10.1112/jlms/s1-10.2.162

    Article  MathSciNet  MATH  Google Scholar 

  215. D. H. Lehmer, “On the factors of \({{2}^{n}} \pm 1\),” Bull. Am. Math. Soc. 53, 164–167 (1947).

    Article  MATH  Google Scholar 

  216. D. H. Lehmer, “Recent discoveries of large primes,” Math. Tables Other Aids Comput. 6, 61 (1952).

    Google Scholar 

  217. D. H. Lehmer, “Two new Mersenne primes,” Math. Tables Other Aids Comput. 7, 72 (1953).

    Google Scholar 

  218. D. H. Lehmer, “Computer technology applied to the theory of numbers,” Math. Assoc. Am. 6, 117–151 (1969).

    MathSciNet  Google Scholar 

  219. D. H. Lehmer, “Multiply perfect numbers,” Ann. Math. 2 (1–4), 103–104 (1900–1901).

    Article  MathSciNet  MATH  Google Scholar 

  220. F. Luca and M. T. Phaovibul, “Amicable pairs with few distinct prime factors,” Int. J. Number Theory 12 (7), 1725–1732 (2016). https://doi.org/10.1142/S1793042116501050

    Article  MathSciNet  MATH  Google Scholar 

  221. F. Luca and C. Pomerance, “The range of the sum-of-proper-divisors function,” Acta Arith. 168, 187–199 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  222. F. Luca and H. te Riele, “\(\phi \) and \(\sigma \): From Euler to Erdős,” Nieuw Arch. Wisk. 12 (1), 31–36 (2011).

    Google Scholar 

  223. F. Luca and H. te Riele, “Aliquot cycles of repdigits,” Integers 12 (1), 129–140 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  224. E. Lucas, “Sur la recherche des grands nombres premiers,” Assoc. Française pour l’Avancement des Sci. 5, 61–68 (1876).

    Google Scholar 

  225. A. R. G. Macdivitt, “The most recently discovered prime number,” Math. Gaz. 63 (426), 268–270 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  226. Th. E. Mason, “On amicable numbers and their generalizations,” Am. Math. Mon. 28, 195–200 (1921). https://doi.org/10.1080/00029890.1921.11986031

    Article  MathSciNet  MATH  Google Scholar 

  227. P. J. McCarthy, “Odd perfect numbers,” Scr. Math. 23, 43–47 (1957).

    MathSciNet  MATH  Google Scholar 

  228. W. L. McDaniel, “On odd multiply perfect numbers,” Boll. Un. Mat. Ital. 2, 185–190 (1970).

    MathSciNet  MATH  Google Scholar 

  229. W. L. McDaniel and P. Hagis, Jr., “Some results concerning the non-existence of odd perfect numbers of the form \({{p}^{\alpha }}{{M}^{{2\beta }}}\),” Fibonacci Q. 13, 25–28 (1975).

    MathSciNet  MATH  Google Scholar 

  230. M. Mercenne, “Cogitata physico mathematica” (1644). https://gallica.bnf.fr/ark:/12148/ bpt6k81531h.r=mersenne.langFR. Accessed November 14, 2020.

  231. M. Mercenne, “Novarum observationum physico mathématicarum” (1647). https://gallica.bnf.fr/ark:/12148/bpt6k815336.r= mersenne.langFR. Accessed November 14, 2020.

  232. D. Minoli, “Issues in nonlinear hyperperfect numbers,” Math. Comput. 34 (150), 639–645 (1980). https://doi.org/10.1090/S0025-5718-1980-0559206-9

    Article  MathSciNet  MATH  Google Scholar 

  233. D. Moews and P. C. Moews, “A search for aliquot cycles below 1010,” Math. Comput. 57, 849–855 (1991).

    MathSciNet  MATH  Google Scholar 

  234. D. Moews and P. C. Moews, “A search for aliquot cycles and amicable pairs,” Math. Comput. 61, 935–938 (1993). https://doi.org/10.1090/S0025-5718-1993-1185249-X

    Article  MathSciNet  MATH  Google Scholar 

  235. D. Moews and P. C. Moews, “A list of amicable pairs below \(2.01 \times {{10}^{{11}}}\),” January 8, 1993. https://xraysgi.ims.uconn.edu:8080/amicable.txt. Accessed November 14, 2020.

  236. D. Moews and P. C. Moews, “A list of the first 5001 amicable pairs,” January 7, 1996. https://xraysgi.ims.uconn.edu:8080/amicable2.txt. Accessed November 14, 2020.

  237. C. G. Moreira and N. C. Saldanha, “Primos de Mersenne (e outros primos muito grandes),” in 22o Colóquio Brasileiro de Matemática (Matemática Pura e Aplicada, Rio de Janeiro, 1999).

  238. L. Murata and C. Pomerance, “On the largest prime factor of a Mersenne number,” in Number Theory, CRM Proc. Lecture Notes (2004), Vol. 36, pp. 209–218. https://doi.org/10.1090/crmp/036/16

  239. W. Narkiewicz, Classical Problems in Number Theory (Polish Scientific, Warszawa, 1986).

    MATH  Google Scholar 

  240. H. M. Nguyen and C. Pomerance, “The reciprocal sum of the amicable numbers,” Math. Comput. 88 (317), 1503–1526 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  241. P. P. Nielsen, “An upper bound for odd perfect numbers,” Integers A 3 (14), 1–9 (2003).

    MathSciNet  MATH  Google Scholar 

  242. P. P. Nielsen, “Odd perfect numbers have at least nine distinct prime factors,” Math. Comput. 76, 2109–2126 (2007). https://doi.org/10.1090/S0025-5718-07-01990-4

    Article  MathSciNet  MATH  Google Scholar 

  243. P. P. Nielsen, “Odd perfect numbers, Diophantine equations, and upper bounds,” Math. Comput. 84 (295), 2549–2567 (2015). https://doi.org/10.1090/S0025-5718-2015-02941-X

    Article  MathSciNet  MATH  Google Scholar 

  244. L. C. Noll and L. Nickel, “The 25th and 26th Mersenne primes,” Math. Comput. 35 (152), 1387–1390 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  245. P. Ochem and M. Rao, “Odd perfect numbers are greater than 101500,” Math. Comput. 81, 1869–1877 (2012). https://doi.org/10.2307/23268069

    Article  MATH  Google Scholar 

  246. P. Ochem and M. Rao, “On the number of prime factors of an odd perfect number,” Math. Comput. 83 (289), 2435–2439 (2014). https://doi.org/10.1090/S0025-5718-2013-02776-7

    Article  MathSciNet  MATH  Google Scholar 

  247. R. Ondrejka, “More on large primes,” J. Recreational Math. 11 (2), 112–113 (1978–1979).

    MathSciNet  MATH  Google Scholar 

  248. R. Ondrejka, “More very large twin primes,” J. Recreational Math. 15 (1), 7 (1982–1983).

    MathSciNet  Google Scholar 

  249. B. N. I. Paganini, Atti della R. Accad. Sc. Torino 2, 362 (1866–1867).

    Google Scholar 

  250. J. Parks, “Amicable pairs and aliquot cycles on average,” Int. J. Number Theory 11 (6), 1751–1790 (2015). https://doi.org/10.1142/S1793042115500761

    Article  MathSciNet  MATH  Google Scholar 

  251. J. M. Pedersen, “Known amicable pairs.” http://amicable.homepage.dk/knwnc2.htm. Accessed November 14, 2020.

  252. J. M. Pedersen, “Various amicable pair lists and statistics.” http://www.vejlehs.dk/staff/jmp/aliquot/apstat.htm. Accessed November 14, 2020.

  253. J. Perrott, “Sur une proposition empirique énoncée au Bulletin,” Bull. Soc. Math. France 17, 155–156 (1889).

    Article  MathSciNet  MATH  Google Scholar 

  254. P. Pollack, “A remark on sociable numbers of odd order,” J. Number Theory 130 (8), 1732–1736 (2010). https://doi.org/10.1016/j.jnt.2010.03.011

    Article  MathSciNet  MATH  Google Scholar 

  255. P. Pollack, “On Dickson’s theorem concerning odd perfect numbers,” Am. Math. Mon. 118 (2), 161–164 (2011). https://doi.org/10.4169/Am.math.monthly.118.02.161

    Article  MathSciNet  MATH  Google Scholar 

  256. P. Pollack, “Powerful amicable numbers,” Colloq. Math. 122 (1), 103–123 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  257. P. Pollack, “Quasi-amicable numbers are rare,” J. Integer Seq. 14, 13 (2011).

    MathSciNet  MATH  Google Scholar 

  258. P. Pollack, “The greatest common divisor of a number and its sum of divisors,” Michigan Math. J. 60, 199–214 (2011). https://doi.org/10.1307/mmj/1301586311

    Article  MathSciNet  MATH  Google Scholar 

  259. P. Pollack, “Finiteness theorems for perfect numbers and their kin,” Am. Math. Mon. 119, 670–681 (2012). Errata: Am. Math. Mon. 120, 482–483 (2013).

    Google Scholar 

  260. P. Pollack, “On relatively prime amicable pairs,” Mosc. J. Comb. Number Theory 5 (1–2), 36–51 (2015).

    MathSciNet  MATH  Google Scholar 

  261. P. Pollack and C. Pomerance, “Prime-perfect numbers,” Integers 12 (6), 1417–1437 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  262. P. Pollack and C. Pomerance, “Paul Erdős and the rise of statistical thinking in elementary number theory,” Erdős Centennial (János Bolyai Math. Soc., Budapest, 2013), pp. 515–533. https://doi.org/10.1007/978-3-642-39286-3_19

    Book  MATH  Google Scholar 

  263. P. Pollack and C. Pomerance, “Some problems of Erdős on the sum-of-divisors function,” Trans. Am. Math. Soc. Ser. B 3, 1–26 (2016). https://doi.org/10.1090/btran/10

    Article  MATH  Google Scholar 

  264. P. Pollack, C. Pomerance, and L. Thompson, “Divisor-sum fibers,” Mathematika 64 (2), 330–342 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  265. P. Pollack and V. Shevelev, “On perfect and near-perfect numbers,” J. Number Theory 132 (12), 3037–3046 (2012). https://doi.org/10.1016/j.jnt.2012.06.008

    Article  MathSciNet  MATH  Google Scholar 

  266. C. Pomerance, “Odd perfect numbers are divisible by at least seven distinct primes,” Acta Arith. 25, 265–300 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  267. C. Pomerance, “Multiply perfect numbers, Mersenne primes, and effective computability,” Math. Ann. 226, 195–206 (1977). https://doi.org/10.1007/BF01362422

    Article  MathSciNet  MATH  Google Scholar 

  268. C. Pomerance, “On the distribution of amicable numbers,” J. Reine Angew. Math. 293–294, 217–222 (1977).

    MathSciNet  MATH  Google Scholar 

  269. C. Pomerance, “On the distribution of amicable numbers II,” J. Reine Angew. Math. 325, 182–188 (1981).

    MathSciNet  MATH  Google Scholar 

  270. C. Pomerance, “On primitive divisors of Mersenne numbers,” Acta Arith. 46 (4), 355–367 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  271. C. Pomerance, “On amicable numbers,” in Analytic Number Theory (Springer, Cham, 2015), pp. 321–327. https://doi.org/10.1007/978-3-319-22240-0_19

    Book  MATH  Google Scholar 

  272. C. Pomerance, “The first function and its iterates,” in Connections in Discrete Mathematics (Cambridge Univ. Press, Cambridge, 2018), pp. 125–138.

    Google Scholar 

  273. C. Pomerance, “The aliquot constant, after Bosma and Kane,” Q. J. Math. 69 (3), 915–930 (2018). https://doi.org/10.1093/qmath/hay005

    Article  MathSciNet  MATH  Google Scholar 

  274. L. Pomey, “Sur les nombres de Fermat et de Mersenne,” Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 16 (3), 135–138 (1924).

    MathSciNet  MATH  Google Scholar 

  275. P. Poulet, Sur les nombres multiparfaits (Association Française, Grenoble, 1925).

    MATH  Google Scholar 

  276. P. Poulet, La chasse aux nombres. I: Parfaits, amiables et extensions (Stevens, Bruxelles, 1929).

  277. P. Poulet, “Forty-three new couples of amicable numbers,” Scr. Math. 14, 77 (1948).

    MATH  Google Scholar 

  278. R. E. Powers, “The tenth perfect number,” Am. Math. Mon. 18 (11), 195–197 (1911). https://doi.org/10.2307/2972574

    Article  MathSciNet  MATH  Google Scholar 

  279. R. E. Powers, “The tenth perfect number,” Bull. Am. Math. Soc. 18 (2), 1 (1912).

    MATH  Google Scholar 

  280. R. E. Powers, “On Mersenne’s numbers,” Proc. London Math. Soc. 13, 39 (1914).

    Google Scholar 

  281. R. E. Powers, “A Mersenne prime,” Bull. Am. Math. Soc. 20 (2), 531 (1914). https://doi.org/10.1090/S0002-9904-1914-02547-9

    Article  MathSciNet  MATH  Google Scholar 

  282. R. E. Powers, “Certain composite Mersenne’s numbers,” Proc. London Math. Soc. 15 (2), 22 (1917).

    MATH  Google Scholar 

  283. R. E. Powers, “Note on a Mersenne number,” Bull. Am. Math. Soc. 40, 883 (1934). https://doi.org/10.1090/S0002-9904-1934-05994-9

    Article  MathSciNet  MATH  Google Scholar 

  284. C. Reid, From Zero to Infinity: What Makes Numbers, 4th ed. (MAA Spectrum, Mathematical Association of America, Washington, DC, 1992).

  285. H. Reidlinger, “Über ungerade mehrfach vollkommene Zahlen,” Österreichische Akad. Wiss. Math. Nat. 192, 237–266 (1983).

    MathSciNet  MATH  Google Scholar 

  286. P. Ribenboim, The Book of Prime Number Records, 2nd ed. (Springer-Verlag, New York, 1989).

    Book  MATH  Google Scholar 

  287. P. Ribenboim, The New Book of Prime Number Records (Springer-Verlag, New York, 1996).

    Book  MATH  Google Scholar 

  288. P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (Springer-Verlag, New York, 2000).

    Book  MATH  Google Scholar 

  289. P. Ribenboim, The Little Book of Bigger Primes, 2nd ed. (Springer, New York, 2004).

    MATH  Google Scholar 

  290. P. Ribenboim, Die Welt der Primzahlen: Geheimnisse und Rekorde, 2nd ed. (Springer, Heidelberg, 2011) [in German].

    Book  MATH  Google Scholar 

  291. G. J. Rieger, “Bemerkung zu einem Ergebnis von Erdős über befreundete Zahlen,” J. Reine Angew. Math. 261, 157–163 (1973).

    MathSciNet  MATH  Google Scholar 

  292. H. J. J. te Riele, “Four large amicable pairs,” Math. Comput. 28, 309–312 (1974). https://doi.org/10.1090/S0025-5718-1974-0330033-8

    Article  MathSciNet  MATH  Google Scholar 

  293. H. J. J. te Riele, “Hyperperfect numbers with three different prime factors,” Math. Comput. 36 (153), 297–298 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  294. H. J. J. te Riele, “Perfect numbers and aliquot sequences: Computational methods in number theory, Part I,” Math. Centre Tracts 154, 141–157 (1982).

    Google Scholar 

  295. H. J. J. te Riele, “New very large amicable pairs,” Number Theory Noordwijkerhout (Springer-Verlag, Berlin, 1984), pp. 210–215. https://doi.org/10.1007/BFb0099454

    Book  Google Scholar 

  296. H. J. J. te Riele, “On generating new amicable pairs from given amicable pairs,” Math. Comput. 42, 219–223 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  297. H. J. J. te Riele, “Rules for constructing hyperperfect numbers,” Fibonacci Q. 22 (1), 50–60 (1984).

    MathSciNet  MATH  Google Scholar 

  298. H. J. J. te Riele, “Computation of all the amicable pairs below 1010,” Math. Comput. 47 (S9–S40), 361–368 (1986).

  299. H. J. J. te Riele, “A new method for finding amicable pairs,” in Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, 50th Anniversary Symposium, Vancouver, BC, 1993 (Am. Math. Soc., Providence, RI, 1994), pp. 577–581.

  300. H. J. J. te Riele, “Grootschalig rekenen in de getaltheorie,” Nieuw Arch. Wisk. 14 (4), 236–243 (2013).

    MathSciNet  Google Scholar 

  301. H. J. J. te Riele, W. Borho, S. Battiato, H. Hoffmann, and E. J. Lee, “Table of amicable pairs between 1010 and 1052,” Technical Report NM-N8603 (Centrum voor Wiskunde en Informatica, Amsterdam, 1986).

  302. H. Riesel, “Några stora primtal,” Elementa 39, 258–260 (1956).

    Google Scholar 

  303. H. Riesel, “A new Mersenne prime,” Math. Tables Aids Comput. 12, 60 (1958).

    Article  MATH  Google Scholar 

  304. H. Riesel, “Mersenne numbers,” Math. Tables Aids Comput.12, 207–213 (1958). https://doi.org/10.2307/2002023

    Article  MathSciNet  MATH  Google Scholar 

  305. H. Riesel, “All factors \(q{{ < 10}^{8}}\) in all Mersenne numbers \({{2}^{p}} - 1\), \(p\) prime \({{ < 10}^{4}}\),” Math. Comput. 16, 478–482 (1962).

    MATH  Google Scholar 

  306. T. Roberts, “On the form of an odd perfect number,” Aust. Math. 35 (4), 244 (2008).

    Article  Google Scholar 

  307. R. M. Robinson, “Mersenne and Fermat numbers,” Proc. Am. Math. Soc. 5, 842–846 (1954). https://doi.org/10.2307/2031878

    Article  MathSciNet  MATH  Google Scholar 

  308. R. M. Robinson, “Some factorizations of numbers of the form \({{2}^{n}} \pm 1\),” Math. Tables Aids Comput. 11, 265–268 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  309. M. I. Rosen, “A proof of the Lucas–Lehmer test,” Am. Math. Mon. 95, 855–856 (1988). https://doi.org/10.1080/00029890.1988.11972101

    Article  MathSciNet  MATH  Google Scholar 

  310. W. W. Rouse Ball, “Mersenne’s numbers,” Messenger Math. 21 (2), 34–40 (1891).

    MATH  Google Scholar 

  311. J. Sándor and B. Crstici, Handbook of Number Theory (Kluwer Academic, Dordrecht, 2004), Vol. 2. https://doi.org/10.1007/1-4020-2547-5

    Book  MATH  Google Scholar 

  312. J. Sándor, D. S. Mitrinović, and B. Crstici, Handbook of Number Theory (Springer, Dordrecht, 2006), Vol. 1.

    MATH  Google Scholar 

  313. M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (Harper Collins, New York, 2003).

    MATH  Google Scholar 

  314. D. Scheffler and R. Ondrejka, “The numerical evaluation of the eighteenth perfect number,” Math. Comput. 14, 199–200 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  315. A. Schinzel and W. Sierpiński, “Sur certaines hypothèses concernant les nombres premiers,” Acta Arith. 4, 185–208 (1958). Erratum: Acta Arith. 5, 259 (1958).

    Google Scholar 

  316. J. L. Selfridge and A. Hurwitz, “Fermat numbers and Mersenne numbers,” Math. Comput. 18, 146–148 (1964). https://doi.org/10.1090/S0025-5718-1964-0159775-8

    Article  MathSciNet  MATH  Google Scholar 

  317. D. Shanks and S. Kravitz, “On the distribution of Mersenne divisors,” Math. Comput. 21, 97–101 (1967). https://doi.org/10.2307/2003474

    Article  MathSciNet  MATH  Google Scholar 

  318. W. Sierpiński, “Les nombres de Mersenne et de Fermat,” Matematiche (Catania) 10, 80–91 (1955).

    MathSciNet  MATH  Google Scholar 

  319. W. Sierpiński, “O liczbach złoḍżonych postaci \({{(2}^{p}} + 1){\text{/}}3\), gdzie \(p\) jest liczbą pierwszą,” Prace Mat. 7, 169–172 (1962).

    MathSciNet  Google Scholar 

  320. W. Sierpiński, Elementary Theory of Numbers, 2nd ed. (PWN–Polish Scientific, Warszawa, 1988).

    MATH  Google Scholar 

  321. V. Sitaramaiah and M. V. Subbarao, “On unitary multiperfect numbers,” Nieuw Arch. Wisk. 16 (1–2), 57–61 (1998).

    MathSciNet  MATH  Google Scholar 

  322. L. Skula, “Prime power divisors of Mersenne numbers and Wieferich primes of higher order,” Integers 19, A19 (2019).

    MathSciNet  MATH  Google Scholar 

  323. D. Slowinski, “Searching for the 27th Mersenne prime,” J. Recreational Math. 11 (4), 258–267 (1978–1979).

    MathSciNet  MATH  Google Scholar 

  324. R. Steuerwald, “Verschärfung einen notwendigen Bedeutung für die Existenz einen ungeraden vollkommenen Zahl,” Bayer. Akad. Wiss. Math. Nat. 2, 69–72 (1937).

    MATH  Google Scholar 

  325. R. Steuerwald, “Ein Satz über natürliche Zahlen mit \(\sigma (N) = 3N\),” Arch. Math. 5, 449–451 (1954). https://doi.org/10.1007/BF0189838910.1007/BF01898389

    Article  MathSciNet  MATH  Google Scholar 

  326. I. Stewart, Professor Stewart’s Incredible Numbers (Basic Books, New York, 2015).

    MATH  Google Scholar 

  327. I. Stewart, Do Dice Play God? The Mathematics of Uncertainty (Basic Books, New York, 2019).

    MATH  Google Scholar 

  328. M. V. Subbarao, “Odd perfect numbers: Some new issues,” Period. Math. Hung. 38 (1–2), 103–109 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  329. M. V. Subbarao, T. J. Cook, R. S. Newberry, and J. M. Weber, “On unitary perfect numbers,” Delta (Waukesha) 3 (1), 22–26 (1972–1973).

  330. M. V. Subbarao and L. J. Warren, “Unitary perfect numbers,” Can. Math. Bull. 9, 147–153 (1966). https://doi.org/10.4153/CMB-1966-018-4

    Article  MathSciNet  MATH  Google Scholar 

  331. D. Suryanarayana, “Super perfect numbers,” Elem. Math. 24, 16–17 (1969).

    MathSciNet  MATH  Google Scholar 

  332. D. Suryanarayana, “Quasiperfect numbers,” Bull. Calcutta Math. Soc. 69, 421–426 (1977).

    MathSciNet  MATH  Google Scholar 

  333. Y. Suzuki, “On amicable tuples,” Ill. J. Math. 62 (1–4), 225–252 (2018). https://doi.org/10.1215/ijm/1552442661

    Article  MathSciNet  MATH  Google Scholar 

  334. J. J. Sylvester, “Sur l’impossibilité de l’existence d’un nombre parfait impair qui ne contient pas au moins \(5\) diviseurs premiers distincts,” C. R. Acad. Sci. Paris 106, 522–526 (1888).

    MATH  Google Scholar 

  335. J. Touchard, “On prime numbers and perfect numbers,” Scr. Math. 19, 35–39 (1953).

    MathSciNet  MATH  Google Scholar 

  336. L. Troupe, “On the number of prime factors of values of the sum-of-proper-divisors function,” J. Number Theory C 150, 120–135 (2015). https://doi.org/10.1016/j.jnt.2014.11.014

    Article  MathSciNet  MATH  Google Scholar 

  337. B. Tuckerman, “The 24th Mersenne prime,” Proc. Nat. Acad. Sci. U.S.A. 68, 2319–2320 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  338. B. Tuckerman, “A search procedure and lower bound for odd perfect numbers,” Math. Comput. 27, 943–949 (1973). Corrigenda: Math. Comput. 28, 887 (1974).

    MathSciNet  Google Scholar 

  339. B. Tuckerman, Corrigendum: “Three new Mersenne primes and a statistical theory,” Math. Comput. 31 (140), 1051 (1977). https://doi.org/10.1090/S0025-5718-1977-0441839-1

    Article  MathSciNet  Google Scholar 

  340. H. S. Uhler, “First proof that the Mersenne number \({{M}_{{157}}}\) is composite,” Proc. Nat. Acad. Sci. U.S.A. 30, 314–316 (1944).

    Article  MathSciNet  MATH  Google Scholar 

  341. H. S. Uhler, “Note on the Mersenne numbers \({{M}_{{157}}}\) and \({{M}_{{167}}}\),” Bull. Am. Math. Soc. 52, 178 (1946).

    Article  MATH  Google Scholar 

  342. H. S. Uhler, “A new result concerning a Mersenne number,” Math. Tables Other Aids Comput. 2, 94 (1946).

    MathSciNet  MATH  Google Scholar 

  343. H. S. Uhler, “On Mersenne’s number \({{M}_{{199}}}\) and Lucas’s sequences,” Bull. Am. Math. Soc. 53, 163–164 (1947).

    Article  MATH  Google Scholar 

  344. H. S. Uhler, “On Mersenne’s number \({{M}_{{227}}}\) and cognate data,” Bull. Am. Math. Soc. 54, 379 (1948).

    Google Scholar 

  345. H. S. Uhler, “On all of Mersenne’s numbers particularly \({{M}_{{193}}}\),” Proc. Nat. Acad. Sci. U.S.A. 34, 102–103 (1948).

    Article  MathSciNet  Google Scholar 

  346. H. S. Uhler, “A brief history of the investigations on Mersenne’s numbers and the latest immense primes,” Scr. Math. 18, 122–131 (1952).

    MathSciNet  MATH  Google Scholar 

  347. H. S. Uhler, “On the 16th and 17th perfect numbers,” Scr. Math. 19, 128–131 (1953).

    MathSciNet  MATH  Google Scholar 

  348. H. S. Uhler, “Full values of the first seventeen perfect numbers,” Scr. Math. 20, 240 (1954).

    MathSciNet  MATH  Google Scholar 

  349. C. H. Viêt, “What’s special about the perfect number 6?” Am. Math. Mon. 128 (1), 87 (2020). https://doi.org/10.1080/00029890.2021.1839304

    Article  MathSciNet  MATH  Google Scholar 

  350. S. S. Wagstaff, Jr., “Divisors of Mersenne numbers,” Math. Comput. 40, 385–397 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  351. S. S. Wagstaff, Jr., “The Cunningham project,” in High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams (Am. Math. Soc., Providence, RI, 2004), pp. 367–378.

  352. S. S. Wagstaff, Jr., The Joy of Factoring (Am. Math. Soc., Providence, RI, 2013).

    Book  MATH  Google Scholar 

  353. A. Walker, “New large amicable pairs” (2004). https://listserv.nodak.edu/scripts/wa.exe?A2=ind0405&L=nmbrthry&F=&S=&P_43. Accessed November 14, 2020.

  354. C. R. Wall, “Bi-unitary perfect numbers,” Proc. Am. Math. Soc. 33, 39–42 (1972). https://doi.org/10.1090/S0002-9939-1972-0289403-9

    Article  MathSciNet  MATH  Google Scholar 

  355. L. J. Warren and H. G. Bray, “On the square-freeness of Fermat and Mersenne numbers,” Pac. J. Math. 22, 563–564 (1967). https://doi.org/10.2140/pjm.1967.22.563

    Article  MathSciNet  MATH  Google Scholar 

  356. J. Watkins, Number Theory: A Historical Approach (Princeton Univ. Press, Princeton, NJ, 2014).

    MATH  Google Scholar 

  357. G. C. Webber, “Non-existence of odd perfect numbers of the form \({{3}^{{2\beta }}}{{p}^{\alpha }}s_{1}^{{2{{\beta }_{1}}}}s_{2}^{{2{{\beta }_{2}}}}s_{3}^{{2{{\beta }_{3}}}}\),” Duke Math. J. 18, 741–749 (1951). https://doi.org/10.1215/S0012-7094-51-01867-410.1215/S0012-7094-51-01867-4

    Article  MathSciNet  Google Scholar 

  358. A. E. Western, “On Lucas’ and Pepin’s tests for the primality of Mersenne’s numbers,” J. London Math. Soc. 7, 130–137 (1932). Corrigenda: J. London Math. Soc. 7 (4), 272 (1932).

    MathSciNet  Google Scholar 

  359. H. C. Williams, Edouard Lucas and Primality Testing (Wiley, New York, 1998).

    MATH  Google Scholar 

  360. H. C. Williams and J. O. Shallit, “Factoring integers before computers,” Math. Comput. 48, 481–531 (1994).

    MathSciNet  MATH  Google Scholar 

  361. E. Wirsing, “Bemerkung zu der Arbeit über vollkommene Zahlen,” Math. Ann. 137, 316–318 (1959). https://doi.org/10.1007/BF01360967

    Article  MathSciNet  MATH  Google Scholar 

  362. H. J. Woodall, “Note on a Mersenne number,” Bull. Am. Math. Soc. 17, 540 (1911).

    Article  MathSciNet  MATH  Google Scholar 

  363. H. J. Woodall, “Mersenne’s numbers,” Manchester Mem. Proc. 56 (1), 1–5 (1912).

    Google Scholar 

  364. G. Woltman, “GIMPS: Mathematics and research strategy.” http://mersenne.org/math.htm. Accessed November 14, 2020.

  365. G. Woltman, “On the discovery of the 38th known Mersenne prime,” Fibonacci Q. 37, 367–370 (1999).

    MathSciNet  MATH  Google Scholar 

  366. G. Woltman and S. Kurowski, “On the discovery of the 45th and 46th known Mersenne primes,” Fibonacci Q. 46–47 (3), 194–197 (2008–2009).

    MathSciNet  MATH  Google Scholar 

  367. T. Yamada, “On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high power of a prime,” Integers 20, A91 (2020).

    MathSciNet  MATH  Google Scholar 

  368. A. Yamagami, “On the numbers of prime factors of square free amicable pairs,” Acta Arith. 177 (2), 153–167 (2017). https://doi.org/10.4064/aa8327-7-2016

    Article  MathSciNet  MATH  Google Scholar 

  369. S. Yates, “Titanic primes,” J. Recreational Math. 16 (4), 250–262 (1983–1984).

    MathSciNet  MATH  Google Scholar 

  370. S. Yates, “Sinkers of the titanics,” J. Recreational Math. 17 (4), 268–274 (1984–1985).

    MathSciNet  MATH  Google Scholar 

  371. S. Yates, “Tracking titanics, in The Lighter Side of Mathematics: Proceedings of Strens Mem. Conf., Calgary 1986 (Math. Assoc. of America, Washington DC, 1993), pp. 349–356.

  372. Y. Pingzhi, “An upper bound for the number of odd multiperfect numbers,” Bull. Aust. Math. Soc. 89 (1), 1–4 (2014). https://doi.org/10.1017/S000497271200113X

    Article  MathSciNet  MATH  Google Scholar 

  373. Y. Pingzhi and Z. Zhongfeng, “Addition to ‘An upper bound for the number of odd multiperfect numbers’,” Bull. Aust. Math. Soc. 89 (1), 5–7 (2014). https://doi.org/10.1017/S0004972713000452

    Article  MathSciNet  MATH  Google Scholar 

  374. J. Zelinsky, “Upper bounds on the second largest prime factor of an odd perfect number,” Int. J. Number Theory 15 (6), 1183–1189 (2019). https://doi.org/10.1142/S1793042119500659

    Article  MathSciNet  MATH  Google Scholar 

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ACKNOWLEDGMENTS

I owe many thanks to Vladimir Khalin, with whom I started this whole affair 15–20 years ago, and to Aleksandr Yurkov, who breathed new life into it. I give special thanks to Sergei Pozdnyakov, who convinced me to write this series of articles. I had very useful discussions with Galina Ivanovna Sinkevich, which influenced the content of the last sections. I am indebted to Boris Kunyavsky, Aleksei Stepanov, and Ilya Shkredov, who read the first draft of this article with great care and suggested a large number of corrections and improvements.

Funding

The study was financially supported by the Russian Foundation for Basic Research within the framework of the scientific project no. 19-29-14141 “Studying the Relationship between Conceptual Mathematical Concepts and Their Digital Representations and Meanings As the Basis for Transformation of School Mathematical Education.”

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    N. A. Vavilov

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Vavilov, N.A. Computers As a Novel Mathematical Reality: III. Mersenne Numbers and Sums of Divisors. Dokl. Math. 107, 173–204 (2023). https://doi.org/10.1134/S1064562423700783

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