Abstract
We recall Charles Babbage’s 1819 criterion for primality, based on simultaneous congruences for binomial coefficients, and extend it to a least-prime-factor test. We also prove a partial converse of his non-primality test, based on a single congruence. Along the way we encounter Bachet, Bernoulli, Bézout, Euler, Fermat, Kummer, Lagrange, Lucas, Vandermonde, Waring, Wilson, Wolstenholme, and several contemporary mathematicians.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
C. Babbage, Demonstration of a theorem relating to prime numbers, Edinburgh Phil. J. 1, 46–49 (1819), http://books.google.com/books?id=KrA-AAAAYAAJ&pg=PA46
C. Babbage, Passages from the Life of a Philosopher, (Longman, Green, Longman, Roberts, & Green, London, 1864), http://djm.cc/library/Passages_Life_of_a_Philosopher_Babbage_edited.pdf
C.G. Bachet, Problèmes plaisants et délectables, qui se font par les nombres, 2nd edn. (Rigaud, Lyon, 1624), http://bsb3.bsb.lrz.de/~db/1008/bsb10081407/images/bsb10081407_00036
W.A. Beyer, Review of [7]. Am. Math. Mon. 86, 66–67 (1979)
B.D. Blackwood, Charles Babbage. In: ed. by D.R. Franceschetti Biographical Encyclopedia of Mathematicians. (Cavendish, New York, 1998), pp. 33–36, http://www.blackwood.org/Babbage.htm
É. Barbin, J. Borowczyk, J.-L. Chabert, A. Djebbar, M. Guillemot, J.-C. Martzloff, A. Michel-Pajus, A History of Algorithms: From the Pebble to the Microchip. ed. by J.-L. Chabert. Trans. by C. Weeks (Springer, Berlin and Heidelberg, 2012)
J.M. Dubbey, The Mathematical Work of Charles Babbage (Cambridge University Press, Cambridge, 1978)
N.J. Fine, Binomial coefficients modulo a prime. Am. Math. Mon. 54, 589–592 (1947)
A. Gardiner, Four problems on prime power divisibility. Am. Math. Mon. 95, 926–931 (1988)
J. Grabiner, Review of From Newton to Hawking: A History of Cambridge University’s Lucasian Professors of Mathematics by K.C. Knox, R. Noakes. Am. Math. Mon. 112, 757–762 (2005)
A. Granville, Arithmetic properties of binomial coefficients I: binomial coefficients modulo prime powers. In: J. Borwein (ed), Organic mathematics (Burnaby, BC, 1995). CMS Conference Proceeding Vol. 20 (American Mathematical Society, Providence, RI, 1997), pp. 253–275, http://www.dms.umontreal.ca/~andrew/Binomial/
R.K. Guy, Unsolved Problems in Number Theory, 3rd edn. (Springer, New York, 2004)
E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. Reine Angew. Math. 44, 93–146 (1852)
J.L. Lagrange, Démonstration d’un théorème nouveau concernant les nombres premiers, Nouv. Mém. Acad. Roy. Sci. Belles-Letters, Berlin 2, 125–137 (1771); available at https://books.google.com/books?id=_-U_AAAAYAAJ&pg=PA125
É. Lucas, Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. France 6, 49–54 (1878), http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1878__6_/BSMF_1878__6__49_1/BSMF_1878__6__49_1.pdf
R.J. McIntosh, On the converse of Wolstenholme’s theorem. Acta Arith. 71, 381–389 (1995)
R. Meštrović, A note on the congruence \(\left(\begin{array}{l}{nd}\\{md}\end{array}\right) \equiv \left(\begin{array}{l}{n}\\{m}\end{array}\right) ({\rm{mod}}\, q)\). Am. Math. Mon. 116, 75–77 (2009)
R. Meštrović, Wolstenholme’s theorem: its generalizations and extensions in the last hundred and fifty years (1862–2011), arXiv:1111.3057 [math.NT] (2011)
R. Meštrović, An extension of Babbage’s criterion for primality, Math. Slovaca 63, 1179–1182 (2013). http://dx.doi.org/10.2478/s12175-013-0164-8
V.H. Moll, Numbers and Functions: From a Classical-Experimental Mathematician’s Point of View. Student Mathematical Library, Vol. 65 (American Mathematical Society, Providence, RI, 2012)
M. Moseley, Irascible Genius: A Life of Charles Babbage, Inventor (Hutchinson, London, 1964)
J.J. O’Connor, E.F. Robertson, Charles Babbage, MacTutor History of Mathematics, http://www-groups.dcs.st-and.ac.uk/history/Biographies/Babbage.html
J.T. O’Donnell, Review of Charles Babbage: Pioneer of the Computer by A. Hymanl. Am. Math. Mon. 92, 522–525 (1985)
C. Pomerance, Divisors of the middle binomial coefficient. Am. Math. Mon. 122, 636–644 (2015)
P. Ribenboim, The Little Book of Bigger Primes (Springer, New York, 2004)
D. Segal, H.W. Brinkmann, E435, Am. Math. Mon. 48, 269–271 (1941)
D. Segal, W. Johnson, E435. Am. Math. Mon. 83, 813 (1976)
Sloane, N.J.A. The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org/ (2017)
V. Trevisan, K. Weber, Testing the converse of Wolstenholme’s theorem. Mat. Contemp. 21, 275–286 (2001)
A.-T. Vandermonde, Mémoire sur des irrationnelles de différens ordres, avec une application au cercle, Mém. Acad. Roy. Sci. Paris (1772), 489–498, http://gallica.bnf.fr/ark:/12148/bpt6k3570q/f79
E. Waring, Meditationes Algebraicae (Cambridge University Press, Cambridge, 1770)
J. Wolstenholme, On certain properties of prime numbers, Q. J. Pure Appl. Math. 5, 35–39 (1862), http://books.google.com/books?id=vL0KAAAAIAAJ&pg=PA35
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Sondow, J. (2017). Extending Babbage’s (Non-)Primality Tests. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-68032-3_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68030-9
Online ISBN: 978-3-319-68032-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)