From: wgd@martigny.ai.mit.edu (Bill Dubuque) Newsgroups: sci.math.numberthy Subject: Re: diophantine equation Date: 30 Nov 98 13:00:32 GMT Giuseppe Melfi wrote: | 2 4 | The only solutions of 1 + x = 2 y | | are (x,y)=(1,1) and (x,y)=(239,13). There is a long proof in an | old paper of Lujengrin. Does somebody knowes a simpler proof ? See the papers reviewed below. There is also much related work which may be located by searching Math Reviews and Zbl for "Ljunggren" (cf. the URLs below). For background see R. K. Guy, Unsolved Problems in Number Theory, Second Edition, 1994, Section D6, and P. Ribenboim, Catalan's Conjecture, 1994, end of part A. -Bill Dubuque ============================================================================== Steiner, Ray; Tzanakis, Nikos Simplifying the solution of Ljunggren's equation X^2 + 1 = 2 Y^4. J. Number Theory 37 (1991), no. 2, 123--132. 11D25 (11J86) ============================================================================== A new proof is given that the equation in the title has only two solutions (1,1) and (293, 13) in positive integers (X,Y). This was originally proved by W. L. Ljunggren [Avh. Norske Vid. Akad. Oslo 1 (1942), no. 5; MR 8,6]. Ljunggren's proof is difficult and complicated and the present authors regard theirs to be "conceptually quite simple". First they reduce the problem to solving the equation x^4 - 12 x^2 y^2 + 16 x y^3 - 4 y^4 = +-1, and then they apply the technique of linear forms in logarithms of units of a suitable quartic field to show that this equation has only the solutions +-(x,y) = (1,3), (1,0), (1,1), (5,2). A deep argument used in the proof is M. Mignotte and M. Waldschmidt's result on a lower bound for linear forms in logarithms of algebraic numbers [Acta Arith. 53 (1989), no. 3, 251--287; MR 90k:11092]. Reviewed by K. Szymiczek Cited in reviews: 95i:11019 http://www.ams.org/mathscinet-getitem?mr=91m:11018 ------------------------------------------------------------------------------ W. Ljunggren solved the title equation in his 1942 paper [Zur Theorie der Gleichung x^2 + 1 = D y^4, Avh. Norske Vid. Akad. Oslo No.5, 1-27 (1942; Zbl. 27, 11)] by a rather complicated argument depending on the structure of units of relative norm -1 in a quadratic extension of a quartic number field. The solutions are (x,y) = (1,1), (239,13) and no others. In the present paper the authors offer an entirely different proof, essentially depending on a deep result of M. Mignotte and M. Waldschmidt [Acta Arith. 53, 251-287 (1989; Zbl. 642.10034)] on lower bounds of linear forms in logarithms of algebraic numbers. It is remarkable that the high-precision calculations involved do not require more than 30 significant decimal digits. [ R.J.Stroeker ] http://www.emis.de:80/cgi-bin/zben/MATH?format=complete&AN=716.11016 ============================================================================== Chen, Jian Hua. A new solution of the Diophantine equation X^2 + 1 = 2 Y^4. J. Number Theory 48 (1994), no. 1, 62--74. 11D25 ============================================================================== The equation (1) x^2 + 1 = 2 y^4 was first solved by W. Ljunggren [Avh. Norske Vid. Akad. Oslo 1942, No. 5, 1-27 (1942; Zbl. 27, 11)] by using Skolem's method. Answering a question of Mordell, R. P. Steiner and N. Tzanakis [J. Number Theory 37, 123-132 (1991; Zbl. 716.11016)] gave a simpler proof by using Baker's method and numerical reduction algorithms. In the present paper the author uses the Thue-Siegel method and Pade approximations to algebraic functions to deduce the surprisingly sharp upper bound |y| < 4.233*10^7*|N|^{1/0.3676} for the solutions of the Thue equation (2) x^4 - 12 x^2 y^2 + 16 x y^3 - 4 y^4 = N. This bound allows equation (2) to be solved for N = 1 from which one obtains the solutions of (1). [ I.Gaal (Debrecen) ] http://www.emis.de:80/cgi-bin/zben/MATH?format=complete&AN=814.11021 ------------------------------------------------------------------------------ In this paper, the old and well-known Diophantine equation of the title is solved by a method rather unusual for such a purpose, namely, by using explicit Pade approximations to algebraic functions. After nine technical lemmas, it is concluded that, if N is a rational integer, then any integer solution (x,y) of (*) X^4 - 12 X^2 Y^2 + 16 X Y^3 - 4 Y^4 = N satisfies |y| < 4.233*10^7*|N|^{1/0.3676}. If N=+-1, this implies a relatively small bound for |y|, and by a direct search all integer solutions of (*) for N=+-1 are obtained; these imply all integer solutions of the equation in the title. In his introduction, the author compares his method of solution to that of R. P. Steiner and the reviewer [J. Number Theory 37 (1991), no. 2, 123--132; MR 91m:11018], which, as he says, "needs a large amount of computation and to know the units in some quartic field". The "large amount of computation" consists in calculating with a precision of no more than 30 decimal digits the roots of a quartic equation and logarithms of some algebraic number, and finding the continued fraction expansion of a real number. The "units in some quartic field" are those of the field generated by the left-hand side of (*). Nowadays, all these things are standard tools of a number of computer packages, which run smoothly even on a PC. Moreover, the solution of Steiner and the reviewer applies a general method of the reviewer and B. M. M. de Weger [J. Number Theory 31 (1989), no. 2, 99--132; MR 90c:11018]. In turn, the author's method of solution, though very interesting---being so unconventional---is probably not of a general character; at least, the author does not claim it to be so. Moreover, the avoidance of the above-mentioned computations is attained at the cost of making ad hoc choices of very many parameters, a fact that prevents the proof from being transparent. The reviewer's views are probably subjective; therefore he strongly recommends to all those interested in the explicit solution of Diophantine equations to look more closely at this interesting paper and make their own comparison with the solution of Steiner and the reviewer. Reviewed by Nikos Tzanakis http://www.ams.org/mathscinet-getitem?mr=95i:11019