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Enumerating solutions to \(p(a)+q(b)=r(c)+s(d)\). (English) Zbl 0960.11055
Let \(p_i(t)\), \(1\leq i\leq 4\), be a polynomial with integral rational coefficients. The author develops a fast method of finding the solutions of the Diophantine equation \(\sum_{i=1}^4 p_i(x_i)= 0\), \(x\in \mathbb{Z}^4\), and applies his algorithm to exhibit the smallest integer representable as a sum of two cubes of (positive) integers in \(k\) different ways for \(k\leq 7\) \((k\leq 5)\), to list 516 solutions of the equation \(x_1^4+ x_2^4=x_3^4+ x_4^4\) (with h.c.f. \((x_1,x_2, x_3,x_4)= 1\)), to give the seven (essentially different) positive solutions with \(x_4\leq 2.1\cdot 10^7\) of the Euler equation \(x_1^4+ x_2^4+ x_3^4= x_4^4\), and to a few other problems.
Reviewer: B.Z.Moroz (Bonn)

MSC:
11Y50 Computer solution of Diophantine equations
11D25 Cubic and quartic Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11P05 Waring’s problem and variants
Software:
heapsort
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References:
[1] Svante Carlsson, Average-case results on Heapsort, BIT 27 (1987), no. 1, 2 – 17. · Zbl 0631.68057 · doi:10.1007/BF01937350 · doi.org
[2] Randy L. Ekl, Equal sums of four seventh powers, Math. Comp. 65 (1996), no. 216, 1755 – 1756. · Zbl 0853.11024
[3] Randy L. Ekl, New results in equal sums of like powers, Math. Comp. 67 (1998), no. 223, 1309 – 1315. · Zbl 0914.11012
[4] Noam D. Elkies, On \?\(^{4}\)+\?\(^{4}\)+\?\(^{4}\)=\?\(^{4}\), Math. Comp. 51 (1988), no. 184, 825 – 835. · Zbl 0698.10010
[5] Robert W. Floyd, Algorithm 245: Treesort3, Communications of the ACM 7 (1964), 701.
[6] Roger E. Frye, Finding \(95800^4+217519^4+414560^4=422481^4\) on the Connection Machine, in [15], 106-116.
[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. · Zbl 0805.11001
[8] D. R. Heath-Brown, The density of zeros of forms for which weak approximation fails, Math. Comp. 59 (1992), no. 200, 613 – 623. · Zbl 0778.11017
[9] Donald E. Knuth, The art of computer programming. Volume 3, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. Sorting and searching; Addison-Wesley Series in Computer Science and Information Processing. · Zbl 0302.68010
[10] Donald E. Knuth, The art of computer programming, volume 3: sorting and searching, second edition, Addison-Wesley, Reading, Massachusetts, 1998.
[11] Leon J. Lander, Thomas R. Parkin, Equal sums of biquadrates, Mathematics of Computation 20 (1966), 450-451. · Zbl 0151.03202
[12] L. J. Lander and T. R. Parkin, A counterexample to Euler’s sum of powers conjecture, Math. Comp. 21 (1967), 101 – 103. · Zbl 0153.06602
[13] L. J. Lander, T. R. Parkin, and J. L. Selfridge, A survey of equal sums of like powers, Math. Comp. 21 (1967), 446 – 459. · Zbl 0149.28803
[14] John Leech, Some solutions of Diophantine equations, Proceedings of the Cambridge Philosophical Society 53 (1957), 778-780. · Zbl 0078.03305
[15] Joanne L. Martin, Stephen F. Lundstrom, Supercomputing ’88: proceedings, volume 2, IEEE Computer Society Press, Silver Spring, Maryland, 1988.
[16] Emmanuel Peyre, Yuri Tschinkel, Tamagawa numbers of diagonal cubic surfaces, numerical evidence, this journal, previous article. · Zbl 0961.14012
[17] E. Rosenstiel, J. A. Dardis, and C. R. Rosenstiel, The four least solutions in distinct positive integers of the Diophantine equation \?=\?³+\?³=\?³+\?³=\?³+\?³=\?³+\?³, Bull. Inst. Math. Appl. 27 (1991), no. 7, 155 – 157. · Zbl 0745.11058
[18] Joseph H. Silverman, Integer points and the rank of Thue elliptic curves, Invent. Math. 66 (1982), no. 3, 395 – 404. · Zbl 0494.14008 · doi:10.1007/BF01389220 · doi.org
[19] Joseph H. Silverman, Integer points on curves of genus 1, J. London Math. Soc. (2) 28 (1983), no. 1, 1 – 7. · Zbl 0487.10015 · doi:10.1112/jlms/s2-28.1.1 · doi.org
[20] Morgan Ward, Euler’s problem on sums of three fourth powers, Duke Mathematical Journal 15 (1948), 827-837. · Zbl 0031.20103
[21] B. Rovan , 15th International Symposium on Mathematical Foundations of Computer Science (MFCS ’90), Elsevier B. V., Amsterdam, 1993. Papers from the symposium held in Bansk√° Bystrica, August 27 – 31, 1990; Theoret. Comput. Sci. 118 (1993), no. 1.
[22] John W. J. Williams, Algorithm 232: Heapsort, Communications of the ACM 7 (1964), 347-348.
[23] Aurel J. Zajta, Solutions of the Diophantine equation \?\(^{4}\)+\?\(^{4}\)=\?\(^{4}\)+\?\(^{4}\), Math. Comp. 41 (1983), no. 164, 635 – 659. · Zbl 0525.10011
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