By Serguei A. Stepanov

Writer S.A. Stepanov completely investigates the present country of the idea of Diophantine equations and its similar tools. Discussions concentrate on mathematics, algebraic-geometric, and logical points of the challenge. Designed for college kids in addition to researchers, the booklet comprises *over 250 excercises* observed by way of tricks, directions, and references. Written in a transparent demeanour, this article doesn't require readers to have specific wisdom of recent equipment of algebraic geometry.

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Van der Poorten. Linear forms in logarithms in p-adic case. In Transcendence Theory: Advances and Applications, edited by A. Baker and D. W. Masser, 29–57. Academic Press, New York. P. Kiss and B. M. Phong. On a function concerning second order recurrences. Ann. Univ. Sci. Budapest. , 21:119–122. N. Robbins. On Fibonacci numbers which are powers. , 16:515–517. R. Steiner. On Fibonacci numbers of the form v 2 + 1. In A Collection of Manuscripts Related to the Fibonacci Sequence, edited by W. E. Hogatt and M.

Nouv. Ann. de Math. (2), 9:469–471, and 10:204–206 (1871). 1878 E. Lucas. Th´eorie des fonctions num´eriques simplement p´eriodiques. Amer. J. , 1:184–240 and 289–321. 1886 A. S. Bang. Taltheoretiske Untersogelser. , Ser. 5, 4:70–80 and 130–137. 1892 K. Zsigmondy. Zur Theorie der Potenzreste. Monatsh. f. , 3:265–284. 1904 G. D. Birkhoﬀ and H. S. Vandiver. On the integral divisors of an − bn . Ann. Math. (2), 5:173–180. 1909 A. Wieferich. Zum letzten Fermatschen Theorem. J. reine u. angew. , 136:293–302.

30: 133–135. 1992 W. L. McDaniel and P. Ribenboim. Squares and double squares in Lucas sequences. C. R. Math. Rep. Acad. Sci. Canada, 14:104–108. 1994 P. Ribenboim. Catalan’s Conjecture. Academic Press, Boston. 1995 P. M. Voutier. Primitive divisors of Lucas and Lehmer sequences. Math. , 64:869–888. 1998a W. L. McDaniel and P. Ribenboim. Square classes in Lucas sequences having odd parameters. J. Nb. , 73:14–23. 1998b W. L. McDaniel and P. Ribenboim. Squares in Lucas sequences having one even parameter.