48 more solutions of Martin Davis’s quaternary quartic equation

Shanks, Daniel; Wagstaff, Samuel S.

doi:10.1090/S0025-5718-1995-1308461-7

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

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.

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Solved and unsolved problems in number theory

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Factoring integers using SIMD sieves