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
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.
 [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
9(𝑥²+7𝑦²)²7(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
(48 #10978)
 [3]
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
(51 #8072)
 [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
(94m:03002b)
 [5]
Daniel
Shanks, Euclid’s primes, Bull. Inst. Combin. Appl.
1 (1991), 33–36. MR 1103634
(92f:11013)
 [6]
Samuel
S. Wagstaff Jr., Computing Euclid’s primes, Bull. Inst.
Combin. Appl. 8 (1993), 23–32. MR 1217356
(94e:11139)
 [7]
Daniel
Shanks and Larry
P. Schmid, Variations on a theorem of Landau.
I, Math. Comp. 20 (1966), 551–569. MR 0210678
(35 #1564), http://dx.doi.org/10.1090/S00255718196602106781
 [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.
 [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)
 [3]
 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)
 [4]
 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.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
11D25,
11Y50
Retrieve articles in all journals
with MSC:
11D25,
11Y50
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
