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

Bernstein, Daniel

doi:10.1090/S0025-5718-00-01219-9

Enumerating solutions to $p(a)+q(b)=r(c)+s(d)$
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by Daniel J. Bernstein PDF

Math. Comp. 70 (2001), 389-394

Abstract:

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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