Small two-variable exponential Diophantine equations

Author:
Robert Styer

Journal:
Math. Comp. **60** (1993), 811-816

MSC:
Primary 11D61

DOI:
https://doi.org/10.1090/S0025-5718-1993-1160277-9

MathSciNet review:
1160277

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Abstract: We examine exponential Diophantine equations of the form . Consider , , , and *b* and *d* from the set of primes 2, 3, 5, 7, 11, and 13. Our work proves that no equation with parameters in these ranges can have solutions with . Our algorithm formalizes and extends a method used by Guy, Lacampagne, and Selfridge in 1987.

**[1]**Alan Baker,*Transcendental number theory*, Cambridge Univ. Press, Cambridge, 1975. MR**0422171 (54:10163)****[2]**John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff, Jr.,*Factorizations of*, Contemp. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1983. MR**715603 (84k:10005)****[3]**B. M. M. De Weger,*Solving exponential Diophantine equations using lattice basis reduction algorithms*, J. Number Theory**26**(1987), 325-367. MR**901244 (88k:11097)****[4]**R. K. Guy, C. B. Lacampagne, and J. L. Selfridge,*Primes at a glance*, Math. Comp.**48**(1987), 183-202. MR**866108 (87m:11008)****[5]**Reese Scott,*On the equation**and*, J. Number Theory (to appear).**[6]**Robert Styer,*A problem of Katai on sums of additive functions*, Acta Sci. Math. (Szeged)**55**(1991), 269-286. MR**1152591 (93d:11004)**

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DOI:
https://doi.org/10.1090/S0025-5718-1993-1160277-9

Article copyright:
© Copyright 1993
American Mathematical Society