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
MathSciNet review:
1308461
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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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 [1]
 Martin Davis, One equation to rule them all, Trans. New York Acad. Sci. (II) 30 (1968), 766773.
 [2]
 Oskar Herrmann, A nontrivial solution of the diophantine equation , Computers in Number Theory, Academic Press, London, 1971, pp. 207212. MR 0332652 (48:10978)
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 Daniel Shanks, Five numbertheoretic algorithms, Proceedings of the Second Manitoba Conference on Numerical Mathematics, University of Manitoba, Winnipeg, 1972, pp. 5170. MR 0371855 (51:8072)
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 Yuri Matiyasevich, Hilbert's Tenth Problem, M. I. T. Press, Cambridge, Mass., 1993. MR 1244324 (94m:03002b)
 [5]
 Daniel Shanks, Euclid's primes, Bull. Inst. Combin. Appl. 1 (1991), 3336. MR 1103634 (92f:11013)
 [6]
 Samuel S. Wagstaff, Jr., Computing Euclid's primes, Bull. Inst. Combin. Appl. 8 (1993), 2332. MR 1217356 (94e:11139)
 [7]
 Daniel Shanks and Larry P. Schmid, Variations on a theorem of Landau, Part I, Math. Comp. 20 (1966), 551569. MR 0210678 (35:1564)
 [8]
 Daniel Shanks, Solved and unsolved problems in number theory, 4th 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, SpringerVerlag, Berlin, 1994, pp. 2839.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199513084617
PII:
S 00255718(1995)13084617
Keywords:
Hilbert's Tenth Problem,
Diophantine equations,
quadratic forms,
factoring integers
Article copyright:
© Copyright 1995 American Mathematical Society
