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Mathematics of Computation

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Enumerating solutions to $p(a)+q(b)=r(c)+s(d)$

Author: Daniel J. Bernstein
Journal: Math. Comp. 70 (2001), 389-394
MSC (2000): Primary 11Y50; Secondary 11D25, 11D41, 11P05, 11Y16
Published electronically: June 12, 2000
MathSciNet review: 1709145
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Let $p,q,r,s$ be polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers $(a,b,c,d)$ satisfying $p(a)+q(b)=r(c)+s(d)$. Numerical results include all small solutions to $a^4+b^4+c^4=d^4$; all small solutions to $a^4+b^4=c^4+d^4$; and the smallest positive integer that can be written in $5$ ways as a sum of two coprime cubes.

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Additional Information

Daniel J. Bernstein
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249) The University of Illinois at Chicago, Chicago, IL 60607–7045

Received by editor(s): July 10, 1998
Received by editor(s) in revised form: January 4, 1999
Published electronically: June 12, 2000
Additional Notes: The author was supported by the National Science Foundation under grant DMS–9600083.
Article copyright: © Copyright 2000 D. J. Bernstein