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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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.
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K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte Math. Phys. 3 (1892), 265284.
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 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
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 J.G. van der Corput, On de Polignac's conjecture, Simon Stevin 27 (1950), 99105. MR 11:714e
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 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36 (1971), 103107. MR 43:3200
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 P. Erdös, On integers of the form and some related problems, Summa Brasil. Math. 2 (1950), 113123. MR 13:437i
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 P.X. Gallagher, Primes and powers of , Invent. Math. 29 (1975), 125142. MR 52:315
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 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
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 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
