On integers not of the form
Author:
ZhiWei Sun
Journal:
Proc. Amer. Math. Soc. 128 (2000), 9971002
MSC (2000):
Primary 11B75; Secondary 11B25, 11P32
Published electronically:
October 27, 1999
MathSciNet review:
1695111
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In 1975 F. Cohen and J.L. Selfridge found a 94digit positive integer which cannot be written as the sum or difference of two prime powers. Following their basic construction and introducing a new method to avoid a bunch of extra congruences, we are able to prove that if then is not of the form where are primes and are nonnegative integers.
 [BV]
G.D. Birkhoff and H.S. Vandiver, On the integral divisors of , Ann. Math. 5 (1904), 173180.
 [CS]
Fred
Cohen and J.
L. Selfridge, Not every number is the sum or
difference of two prime powers, Math. Comp.
29 (1975), 79–81.
Collection of articles dedicated to Derrick Henry Lehmer on the occasion of
his seventieth birthday. MR 0376583
(51 #12758), http://dx.doi.org/10.1090/S00255718197503765830
 [Co]
J.
G. van der Corput, On de Polignac’s conjecture, Simon
Stevin 27 (1950), 99–105 (Dutch). MR 0035298
(11,714e)
 [Cr]
Roger
Crocker, On the sum of a prime and of two powers of two,
Pacific J. Math. 36 (1971), 103–107. MR 0277467
(43 #3200)
 [E]
P.
Erdös, On integers of the form 2^{𝑘}+𝑝 and
some related problems, Summa Brasil. Math. 2 (1950),
113–123. MR 0044558
(13,437i)
 [Ga]
P.
X. Gallagher, Primes and powers of 2, Invent. Math.
29 (1975), no. 2, 125–142. MR 0379410
(52 #315)
 [Gu]
Richard
K. Guy, Unsolved problems in number theory, 2nd ed., Problem
Books in Mathematics, SpringerVerlag, New York, 1994. Unsolved Problems in
Intuitive Mathematics, I. MR 1299330
(96e:11002)
 [GS]
A. Granville and K. Soundararajan, A binary additive problem of Erdös and the order of , Ramanujan J. 2 (1998), 283298. CMP 99:01
 [P]
A. de Polignac, Recherches nouvelles sur les nombres premiers, C. R. Acad. Sci. Paris Math. 29 (1849), 397401, 738739.
 [Ri]
Denis
Richard, All arithmetical sets of powers of primes are firstorder
definable in terms of the successor function and the coprimeness
predicate, Discrete Math. 53 (1985), 221–247
(English, with French summary). Special volume on ordered sets and their
applications (L’Arbresle, 1982). MR 786492
(86h:03103), http://dx.doi.org/10.1016/0012365X(85)90144X
 [Ro]
N.P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 57 (1934), 668678.
 [Si]
W.
Sierpiński, Elementary theory of numbers, 2nd ed.,
NorthHolland Mathematical Library, vol. 31, NorthHolland Publishing
Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw, 1988.
Edited and with a preface by Andrzej Schinzel. MR 930670
(89f:11003)
 [Su]
ZhiWei Sun, On prime divisors of integers and , to appear.
 [VM]
M.V. VassilevMissana, Note on `extraordinary primes', Notes Number Theory Discrete Math. 1 (1995), 111113. MR 97g:11004.
 [Z]
K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte Math. Phys. 3 (1892), 265284.
 [BV]
 G.D. Birkhoff and H.S. Vandiver, On the integral divisors of , Ann. Math. 5 (1904), 173180.
 [CS]
 F. Cohen and J.L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), 7981. MR 51:12758
 [Co]
 J.G. van der Corput, On de Polignac's conjecture, Simon Stevin 27 (1950), 99105. MR 11:714e
 [Cr]
 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36 (1971), 103107. MR 43:3200
 [E]
 P. Erdös, On integers of the form and some related problems, Summa Brasil. Math. 2 (1950), 113123. MR 13:437i
 [Ga]
 P.X. Gallagher, Primes and powers of , Invent. Math. 29 (1975), 125142. MR 52:315
 [Gu]
 R.K. Guy, Unsolved Problems in Number Theory (2nd ed.), SpringerVerlag, New York, 1994, sections A19,B21,F13. MR 96e:11002
 [GS]
 A. Granville and K. Soundararajan, A binary additive problem of Erdös and the order of , Ramanujan J. 2 (1998), 283298. CMP 99:01
 [P]
 A. de Polignac, Recherches nouvelles sur les nombres premiers, C. R. Acad. Sci. Paris Math. 29 (1849), 397401, 738739.
 [Ri]
 D. Richard, All arithmetical sets of powers of primes are firstorder definable in terms of the successor function and the coprimeness predicate, Discrete Math. 53 (1985), 221247. MR 86h:03103
 [Ro]
 N.P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann. 57 (1934), 668678.
 [Si]
 W. Sierpi\'{n}ski, Elementary Theory of Numbers, PWNPolish Scientific Publishers, NorthHolland, Amsterdam, 1987, pp. 445448. MR 89f:11003
 [Su]
 ZhiWei Sun, On prime divisors of integers and , to appear.
 [VM]
 M.V. VassilevMissana, Note on `extraordinary primes', Notes Number Theory Discrete Math. 1 (1995), 111113. MR 97g:11004.
 [Z]
 K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte Math. Phys. 3 (1892), 265284.
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Additional Information
ZhiWei Sun
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email:
zwsun@netra.nju.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002993999055021
PII:
S 00029939(99)055021
Received by editor(s):
June 16, 1998
Published electronically:
October 27, 1999
Additional Notes:
This research was supported by the National Natural Science Foundation of the People’s Republic of China and the Returnfromabroad Foundation of the Chinese Educational Committee
Communicated by:
David E. Rohrlich
Article copyright:
© Copyright 2000
American Mathematical Society
