48 more solutions of Martin Davis’s quaternary quartic equation
Authors:
Daniel Shanks and Samuel S. Wagstaff
Journal:
Math. Comp. 64 (1995), 17171731
MSC:
Primary 11D25; Secondary 11Y50
DOI:
https://doi.org/10.1090/S00255718199513084617
MathSciNet review:
1308461
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: We find 48 more solutions to a Diophantine equation investigated by Martin Davis. Before our work, only two solutions were known. Construction of the new solutions required the factorization of several large integers. Because the equation relates to Hilbert’s Tenth Problem it is desirable to know if it has only finitely many solutions. An elaborate argument is given for the conjecture that the equation has infinitely many solutions in integers.

Martin Davis, One equation to rule them all, Trans. New York Acad. Sci. (II) 30 (1968), 766773.
 Oskar Herrmann, A nontrivial solution of the Diophantine equation $9(x^{2}+7y^{2})^{2}7(u^{2}+7v^{2})^{2}=2$, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 207–212. MR 0332652
 Daniel Shanks, Five numbertheoretic algorithms, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. Congressus Numerantium, No. VII. MR 0371855
 Yu. V. Matiyasevich, Desyataya problema Gil′berta, Matematicheskaya Logika i Osnovaniya Matematiki [Monographs in Mathematical Logic and Foundations of Mathematics], vol. 26, VO “Nauka”, Moscow, 1993 (Russian, with Russian summary). MR 1247235
 Daniel Shanks, Euclid’s primes, Bull. Inst. Combin. Appl. 1 (1991), 33–36. MR 1103634
 Samuel S. Wagstaff Jr., Computing Euclid’s primes, Bull. Inst. Combin. Appl. 8 (1993), 23–32. MR 1217356
 Daniel Shanks and Larry P. Schmid, Variations on a theorem of Landau. I, Math. Comp. 20 (1966), 551–569. MR 210678, DOI https://doi.org/10.1090/S00255718196602106781 Daniel Shanks, Solved and unsolved problems in number theory, 4th ed., Chelsea, New York, 1993. B. Dixon and A. K. Lenstra, Factoring integers using SIMD sieves, Advances in Cryptology, Eurocrypt ’93, Lecture Notes in Comput. Sci., vol. 765, SpringerVerlag, Berlin, 1994, pp. 2839.
Retrieve articles in Mathematics of Computation with MSC: 11D25, 11Y50
Retrieve articles in all journals with MSC: 11D25, 11Y50
Additional Information
Keywords:
Hilbert’s Tenth Problem,
Diophantine equations,
quadratic forms,
factoring integers
Article copyright:
© Copyright 1995
American Mathematical Society