48 more solutions of Martin Davis's quaternary quartic equation

Authors:
Daniel Shanks and Samuel S. Wagstaff

Journal:
Math. Comp. **64** (1995), 1717-1731

MSC:
Primary 11D25; Secondary 11Y50

DOI:
https://doi.org/10.1090/S0025-5718-1995-1308461-7

MathSciNet review:
1308461

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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.

**[1]**Martin Davis,*One equation to rule them all*, Trans. New York Acad. Sci. (II)**30**(1968), 766-773.**[2]**Oskar Herrmann,*A non-trivial solution of the Diophantine equation 9(𝑥²+7𝑦²)²-7(𝑢²+7𝑣²)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****[3]**Daniel Shanks,*Five number-theoretic 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****[4]**Yuri V. Matiyasevich,*Hilbert’s tenth problem*, Foundations of Computing Series, MIT Press, Cambridge, MA, 1993. Translated from the 1993 Russian original by the author; With a foreword by Martin Davis. MR**1244324****[5]**Daniel Shanks,*Euclid’s primes*, Bull. Inst. Combin. Appl.**1**(1991), 33–36. MR**1103634****[6]**Samuel S. Wagstaff Jr.,*Computing Euclid’s primes*, Bull. Inst. Combin. Appl.**8**(1993), 23–32. MR**1217356****[7]**Daniel Shanks and Larry P. Schmid,*Variations on a theorem of Landau. I*, Math. Comp.**20**(1966), 551–569. MR**0210678**, https://doi.org/10.1090/S0025-5718-1966-0210678-1**[8]**Daniel Shanks,*Solved and unsolved problems in number theory*, 4*th ed.*, Chelsea, New York, 1993.**[9]**B. Dixon and A. K. Lenstra,*Factoring integers using SIMD sieves*, Advances in Cryptology, Eurocrypt '93, Lecture Notes in Comput. Sci., vol. 765, Springer-Verlag, Berlin, 1994, pp. 28-39.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1995-1308461-7

Keywords:
Hilbert's Tenth Problem,
Diophantine equations,
quadratic forms,
factoring integers

Article copyright:
© Copyright 1995
American Mathematical Society