Enumerating solutions to 𝑝(π‘Ž)+π‘ž(𝑏)=π‘Ÿ(𝑐)+𝑠(𝑑)

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Abstract
Letp,q,r,sp,q,r,sbe polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers(a,b,c,d)(a,b,c,d)satisfyingp(a)+q(b)=r(c)+s(d)p(a)+q(b)=r(c)+s(d). Numerical results include all small solutions toa4+b4+c4=d4a^4+b^4+c^4=d^4; all small solutions toa4+b4=c4+d4a^4+b^4=c^4+d^4; and the smallest positive integer that can be written in55ways as a sum of two coprime cubes.

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