On integers of the form $k2^{n}+1$
HTML articles powered by AMS MathViewer
- by Yong-Gao Chen PDF
- Proc. Amer. Math. Soc. 129 (2001), 355-361 Request permission
Abstract:
In this paper we show that the set of positive odd integers $k$ such that $k2^{n} +1$ has at least three distinct prime factors for all positive integers $n$ has positive lower asymptotic density.References
- Robert Baillie, New primes of the form $k\cdot 2^{n}+1$, Math. Comp. 33 (1979), no. 148, 1333–1336. MR 537979, DOI 10.1090/S0025-5718-1979-0537979-0
- Robert Baillie, G. Cormack, and H. C. Williams, The problem of Sierpiński concerning $k\cdot 2^{n}+1$, Math. Comp. 37 (1981), no. 155, 229–231. MR 616376, DOI 10.1090/S0025-5718-1981-0616376-2
- A. Baker, The theory of linear forms in logarithms, Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976) Academic Press, London, 1977, pp. 1–27. MR 0498417
- Wieb Bosma, Explicit primality criteria for $h\cdot 2^k\pm 1$, Math. Comp. 61 (1993), no. 203, 97–109, S7–S9. MR 1197510, DOI 10.1090/S0025-5718-1993-1197510-3
- D. A. Buell and J. Young, Some large primes and the Sierpiński problem, SRC Technical Report 88-004, Supercomputing Research Center, Lanham MD, 1988.
- Y. G. Chen, On integers of the form $2^{n} \pm p_{1}^{\alpha _{1} } \cdots p_{r}^{\alpha _{r} }$, Proc. Amer. Math. Soc. (to appear).
- S. L. G. Choi, Covering the set of integers by congruence classes of distinct moduli, Math. Comp. 25 (1971), 885–895. MR 297692, DOI 10.1090/S0025-5718-1971-0297692-7
- Wilfrid Keller, Table errata: “Some very large primes of the form $k\cdot 2^{m}+1$” [Math. Comp. 35 (1980), no. 152, 1419–1421; MR 81i:10011] by G. V. Cormack and H. C. Williams, Math. Comp. 38 (1982), no. 157, 335. MR 637312, DOI 10.1090/S0025-5718-1982-0637312-X
- P. Erdős and A. M. Odlyzko, On the density of odd integers of the form $(p-1)2^{-n}$ and related questions, J. Number Theory 11 (1979), no. 2, 257–263. MR 535395, DOI 10.1016/0022-314X(79)90043-X
- Richard K. Guy, Unsolved problems in number theory, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR 1299330, DOI 10.1007/978-1-4899-3585-4
- G. Jaeschke, Corrigendum: “On the smallest $k$ such that all $k\cdot 2^n+1$ are composite” [Math. Comp. 40 (1983), no. 161, 381–384; MR0679453 (84k:10006)], Math. Comp. 45 (1985), no. 172, 637. MR 804951, DOI 10.1090/S0025-5718-1985-0804951-5
- Wilfred Keller, Corrigendum: “Primes of the form $n!\pm 1$ and $2\cdot 3\cdot 5\cdots p\pm 1$” [Math. Comp. 38 (1982), no. 158, 639–643; MR0645679 (83c:10006)] by J. P. Buhler, R. E. Crandall and M. A. Penk, Math. Comp. 40 (1983), no. 162, 727. MR 689486, DOI 10.1090/S0025-5718-1983-0689486-3
- K. Mahler, On the fractional parts of the powers of a rational number. II, Mathematika 4 (1957), 122–124. MR 93509, DOI 10.1112/S0025579300001170
- D. Ridout, Rational approximations to algebraic numbers, Mathematika 4 (1957), 125–131. MR 93508, DOI 10.1112/S0025579300001182
- Raphael M. Robinson, A report on primes of the form $k\cdot 2^{n}+1$ and on factors of Fermat numbers, Proc. Amer. Math. Soc. 9 (1958), 673–681. MR 96614, DOI 10.1090/S0002-9939-1958-0096614-7
- J. L. Selfridge, Solution of problem 4995, Amer. Math. Monthly 70(1963), 101.
- W. Sierpiński, Sur un problème concernant les nombres $k\cdot 2^{n} +1$, Elem. Math. 15(1960), 73-74; MR 22# 7983, corrigendum, 17(1962), 85.
- R. G. Stanton and H. C. Williams, Further results on covering of the integer $1+k2^{n}$ by primes, Combinatorial Math. VIII, Lecture Notes in Math. 884, Springer-Verlag, Berlin, New York, 1980, 107-114.
- Kun Rui Yu, Linear forms in $p$-adic logarithms. III, Compositio Math. 91 (1994), no. 3, 241–276. MR 1273651, DOI 10.1001/archderm.1962.01590010023003
Additional Information
- Yong-Gao Chen
- Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
- MR Author ID: 304097
- Email: ygchen@pine.ninu.edu.cn
- Received by editor(s): April 29, 1999
- Published electronically: August 28, 2000
- Additional Notes: This research was supported by the Fok Ying Tung Education Foundation and the National Natural Science Foundation of China, Grant No 19701015
- Communicated by: David E. Rohrlich
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 355-361
- MSC (2000): Primary 11A07, 11B25
- DOI: https://doi.org/10.1090/S0002-9939-00-05916-5
- MathSciNet review: 1800230