Skip to main content
SpringerLink
Log in

Sums of primes and quadratic linear recurrence sequences

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let U be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on U in terms of its distribution modulo d, d = 1, 2, …, under which the set of positive integers expressible by the sum of a prime number and an element of U has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form \(p + \left\lfloor {(2 + \sqrt 3 )^n } \right\rfloor \), where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be “too small” for most prime numbers p.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bell, J. P., Gerhold, S.: On the positivity set of a linear recurrence sequence. Israel J. Math., 157, 333–345 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Catlin, P. A.: A lower bound for the period of the Fibonacci series modulo M. Fibonacci Quart., 12, 349–350 (1974)

    MathSciNet  MATH  Google Scholar 

  3. Chen, J. R.: On the Goldbach’s problem and the sieve methods. Sci. Sinica, 21, 701–739 (1978)

    MathSciNet  MATH  Google Scholar 

  4. Chen, Y. G.: On integers of the form \(2^k \pm p_1^{\alpha _1 } p_2^{\alpha _2 } \cdots p_r^{\alpha _r } \). Proc. Amer. Math. Soc., 128, 1613–1616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Y. G., Sun, X. G.: On Romanoff’s constant. J. Number Theory, 106, 275–284 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. de Polignac, A.: Réchèrches nouvelle sur les nombres premieres. C. R. Acad. Sci. Paris, 29, 397–401; Rectification 738–739 (1849)

    Google Scholar 

  7. Enoch Lee, K. S.: On the sum of a prime and a Fibonacci number. Int. J. Number Theory, 6, 1669–1676 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Erdős, P.: On integers of the form 2k + p and some related problems. Summa Brasil. Math., 2, 113–123 (1950)

    MathSciNet  Google Scholar 

  9. Everest, G., van der Poorten, A., Shparlinski, I., et al.: Recurrence Sequences, Mathematical Surveys and Monographs 104, American Mathematical Society, Providence, RI, 2003

    Google Scholar 

  10. Fainleib, A. S., Orazov, A.: Binary additive problems with exponential function. Litovsk. Mat. Sb., 18, 187–198 (1978)

    MathSciNet  MATH  Google Scholar 

  11. Habsieger, L., Roblot, X. F.: On integers of the form p + 2k. Acta Arith., 122, 45–50 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hooley, C.: On Artin’s conjecture. J. Reine Angew Math., 225, 209–220 (1967)

    MathSciNet  MATH  Google Scholar 

  13. Konyagin, S. V., Pomerance, C., Shparlinski, I. E.: On the distribution of pseudopowers. Canad. J. Math., 62, 582–594 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, H., Pan, H.: The Romanoff theorem revisited. Acta Arith., 135, 137–142 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lü, G.: On Romanoff’s constant and its generalized problem. Adv. Math. (China), 36, 94–100 (2007)

    Google Scholar 

  16. Niederreiter, H., Schinzel, A., Somer, L.: Maximal frequencies of elements in second-order linear recurring sequences over a finite field. Elem. Math., 46, 139–143 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Pappalardi, F.: On the order of finitely generated subgroups of Q* (mod p) and divisors of p-1. J. Number Theory, 57, 207–222 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pintz, J.: A note on Romanov’s constant. Acta Math. Hungar., 112, 1–14 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ram Murty, M.: Artin’s conjecture for primitive roots. Math. Intell., 10, 59–67 (1988)

    Article  MATH  Google Scholar 

  20. Romanoff, N. P.: Über einige Sätze der additiven Zahlentheorie. Math. Ann., 57, 668–678 (1934)

    Article  MathSciNet  Google Scholar 

  21. Roskam, H.: Artin’s primitive root conjecture for quadratic fields. J. Théor. des Nombres de Bordeaux, 14, 287–324 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Roskam, H.: Prime divisors of linear recurrences and Artin’s primitive root conjecture for number fields. J. Théor. des Nombres de Bordeaux, 13, 303–314 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schinzel, A.: Special Lucas sequences, including the Fibonacci sequence, modulo a prime. In: A Tribute to Paul Erdős (Baker, A., Bolobás, B., Hajnal, A., eds.), CUP, Cambridge, 1990, 349–357

    Chapter  Google Scholar 

  24. Somer, L.: Distribution of residues of certain second-order linear recurrences modulo p. In: Applications of Fibonacci Numbers (Pisa, 1988) (Bergum, G.E., Philippou, A.N., Horadam, A.F., eds.), Kluwer Acad. Publ., Dordrecht, 1990, 311–324

    Chapter  Google Scholar 

  25. Somer, L.: Primes having an incomplete system of residues for a class of second-order recurrences. In: Applications of Fibonacci Numbers (San Jose, CA, 1986), (Philippou, A. N., Horadam, A. F., Bergum, G. E., eds.), Kluwer Acad. Publ., Dordrecht, 1988, 113–141

    Chapter  Google Scholar 

  26. Sun, X. G.: On the density of integers of the form 2k +p in arithmetic progressions. Acta Math. Sin., Engl. Ser., 26, 155–160 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun, Z. W.: Mixed sums of primes and other terms. In: Additive Number Theory, Festschrift in honor of the sixtieth birthday of M.B. Nathanson (Chudnovsky, D., Chudnovsky, G., eds.), Springer, New York, 2010, 341–353

    Google Scholar 

  28. van der Corput, J. G.: On de Polignac’s conjecture (in Duch). Simon Stevin, 27, 99–105 (1950)

    MathSciNet  MATH  Google Scholar 

  29. Wrench, J. W.: Evaluation of Artin’s constant and the twin-prime constant. Math. Comp., 15, 396–398 (1961)

    MathSciNet  MATH  Google Scholar 

  30. Wu, D. H.: An improvement of J. R. Chen’s theorem (in Chinese). Shanghai Keji Daxue Xuebao, 94–99 (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius, LT-03225, Lithuania

    Artūras Dubickas

Authors
  1. Artūras Dubickas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Artūras Dubickas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dubickas, A. Sums of primes and quadratic linear recurrence sequences. Acta. Math. Sin.-English Ser. 29, 2251–2260 (2013). https://doi.org/10.1007/s10114-013-1059-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-013-1059-x

Keywords

MR(2010) Subject Classification

Search

Navigation