Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers
HTML articles powered by AMS MathViewer
- by Yong-Gao Chen;
- Math. Comp. 74 (2005), 1025-1031
- DOI: https://doi.org/10.1090/S0025-5718-04-01674-6
- Published electronically: July 20, 2004
- PDF | Request permission
Abstract:
In this paper, we prove that there is an arithmetic progression of positive odd numbers for each term $M$ of which none of five consecutive odd numbers $M, M-2, M-4, M-6$ and $M-8$ can be expressed in the form $2^n \pm p^\alpha$, where $p$ is a prime and $n, \alpha$ are nonnegative integers.References
- A. S. Bang, Taltheoretiske Unders$\phi$gelser, Tidsskrift for Mat. (5), 4(1886), 70-80, 130-137.
- G. D. Birkhoff and H. S. Vandiver, On the integral divisors of $a^{n}-b^{n}$, Ann. Math. 5(1904), 173-180.
- Yong-Gao Chen, On integers of the form $2^k\pm p^{\alpha _1}_1p^{\alpha _2}_2\cdots p^{\alpha _r}_r$, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1613–1616. MR 1695159, DOI 10.1090/S0002-9939-99-05482-9
- Yong-Gao Chen, On integers of the form $k2^n+1$, Proc. Amer. Math. Soc. 129 (2001), no. 2, 355–361. MR 1800230, DOI 10.1090/S0002-9939-00-05916-5
- Yong-Gao Chen, On integers of the forms $k-2^n$ and $k2^n+1$, J. Number Theory 89 (2001), no. 1, 121–125. MR 1838707, DOI 10.1006/jnth.2001.2640
- Yong-Gao Chen, On integers of the forms $k^r-2^n$ and $k^r2^n+1$, J. Number Theory 98 (2003), no. 2, 310–319. MR 1955419, DOI 10.1016/S0022-314X(02)00051-3
- Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comp. 29 (1975), 79–81. MR 376583, DOI 10.1090/S0025-5718-1975-0376583-0
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- 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, On the smallest $k$ such that all $k\cdot 2^{n}+1$ are composite, Math. Comp. 40 (1983), no. 161, 381–384. MR 679453, DOI 10.1090/S0025-5718-1983-0679453-8
- N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 57(1934), 668-678.
- R. G. Stanton, Further results on covering integers of the form $1+k2^{n}$ by primes, Combinatorial mathematics, VIII (Geelong, 1980) Lecture Notes in Math., vol. 884, Springer, Berlin, 1981, pp. 107–114. MR 641240
- K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. 3(1892), 265-284.
Bibliographic Information
- Yong-Gao Chen
- Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, Peoples Republic of China
- MR Author ID: 304097
- Email: ygchen@pine.njnu.edu.cn
- Received by editor(s): January 2, 2003
- Received by editor(s) in revised form: October 2, 2003
- Published electronically: July 20, 2004
- Additional Notes: Supported by the National Natural Science Foundation of China, Grant No. 10171046 and the Teaching and Research Award Program for Outstanding Young Teachers in Nanjing Normal University
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 1025-1031
- MSC (2000): Primary 11A07, 11B25
- DOI: https://doi.org/10.1090/S0025-5718-04-01674-6
- MathSciNet review: 2114663