On integers of the form

Author:
Yong-Gao Chen

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1613-1616

MSC (2000):
Primary 11A07, 11B25

DOI:
https://doi.org/10.1090/S0002-9939-99-05482-9

Published electronically:
November 23, 1999

MathSciNet review:
1695159

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove that the set of positive odd integers which have no representation of the form , where , are distinct odd primes and are nonnegative integers, has positive lower asymptotic density in the set of all positive odd integers.

**1.**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****2.**A. S. Bang,*Taltheoretiske Undersgelser*, Tidsskrift for Mat. (5), 4(1886), 70-80, 130-137.**3.**G. D. Birkhoff and H. S. Vandiver,*On the integral divisors of*, Ann. Math. 5(1904), 173-180.**4.**S. L. G. Choi,*Covering the set of integers by congruence classes of distinct moduli*, Math. Comp.**25**(1971), 885–895. MR**297692**, https://doi.org/10.1090/S0025-5718-1971-0297692-7**5.**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**, https://doi.org/10.1090/S0025-5718-1975-0376583-0**6.**P. Erd\H{o}s,*On integers of the form and some related problems*, Summa Brasil. Math. 2(1947-51), 113-123. MR**13:437i****7.**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****8.**N. P. Romanoff,*Über einige Sätze der additiven Zahlentheorie*, Math. Ann. 57(1934), 668-678.**9.**K. Zsigmondy,*Zur Theorie der Potenzreste*, Monatsh. Math. 3(1892), 265-284.

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Additional Information

**Yong-Gao Chen**

Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China

Email:
ygchen@pine.njnu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-99-05482-9

Received by editor(s):
July 20, 1998

Published electronically:
November 23, 1999

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

Article copyright:
© Copyright 2000
American Mathematical Society