A finiteness theorem for a class

of exponential congruences

Authors:
Marian Vâjâitu and Alexandru Zaharescu

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2225-2232

MSC (1991):
Primary 11A07

Published electronically:
April 9, 1999

MathSciNet review:
1486757

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For given elements and belonging to the ring of integers of a number field we consider the set of all tuples in for which divides for any and prove under some reasonable assumptions that the set of solutions is finite.

**1.**Richard K. Guy,*Unsolved problems in number theory*, Unsolved Problems in Intuitive Mathematics, vol. 1, Springer-Verlag, New York-Berlin, 1981. Problem Books in Mathematics. MR**656313**

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****2.**C. Pomerance, Amer. Math. Monthly**84**(1977), 59-60.**3.**S. Ramanujan,*Highly composite numbers*, Proc. London Math. Soc. (2)**14**(1915).**4.**A. Schinzel,*On primitive prime factors of 𝑎ⁿ-𝑏ⁿ*, Proc. Cambridge Philos. Soc.**58**(1962), 555–562. MR**0143728****5.**Qi Sun and Ming Zhi Zhang,*Pairs where 2^{𝑎}-2^{𝑏} divides 𝑛^{𝑎}-𝑛^{𝑏} for all 𝑛*, Proc. Amer. Math. Soc.**93**(1985), no. 2, 218–220. MR**770523**, 10.1090/S0002-9939-1985-0770523-6**6.**B. Velez, Amer. Math. Monthly**83**(1976), 288-289.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
11A07

Retrieve articles in all journals with MSC (1991): 11A07

Additional Information

**Marian Vâjâitu**

Affiliation:
Institute of Mathematics of the Romanian Academy, P.O.Box 1-764, 70700 Bucharest, Romania

Email:
mvajaitu@stoilow.imar.ro

**Alexandru Zaharescu**

Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139

Address at time of publication:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6

Email:
azah@math.mit.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04822-4

Received by editor(s):
October 17, 1995

Received by editor(s) in revised form:
May 20, 1997, and October 28, 1997

Published electronically:
April 9, 1999

Communicated by:
William W. Adams

Article copyright:
© Copyright 1999
American Mathematical Society