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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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48 more solutions of Martin Davis’s quaternary quartic equation
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by Daniel Shanks and Samuel S. Wagstaff PDF
Math. Comp. 64 (1995), 1717-1731 Request permission


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), 766-773.
  • Oskar Herrmann, A non-trivial 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 number-theoretic algorithms, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Congressus Numerantium, No. VII, Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. 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 10.1090/S0025-5718-1966-0210678-1
  • 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, Springer-Verlag, Berlin, 1994, pp. 28-39.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1717-1731
  • MSC: Primary 11D25; Secondary 11Y50
  • DOI:
  • MathSciNet review: 1308461