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Enumerating solutions to $$p(a)+q(b)=r(c)+s(d)$$. (English) Zbl 0960.11055
Let $$p_i(t)$$, $$1\leq i\leq 4$$, be a polynomial with integral rational coefficients. The author develops a fast method of finding the solutions of the Diophantine equation $$\sum_{i=1}^4 p_i(x_i)= 0$$, $$x\in \mathbb{Z}^4$$, and applies his algorithm to exhibit the smallest integer representable as a sum of two cubes of (positive) integers in $$k$$ different ways for $$k\leq 7$$ $$(k\leq 5)$$, to list 516 solutions of the equation $$x_1^4+ x_2^4=x_3^4+ x_4^4$$ (with h.c.f. $$(x_1,x_2, x_3,x_4)= 1$$), to give the seven (essentially different) positive solutions with $$x_4\leq 2.1\cdot 10^7$$ of the Euler equation $$x_1^4+ x_2^4+ x_3^4= x_4^4$$, and to a few other problems.
Reviewer: B.Z.Moroz (Bonn)

##### MSC:
 11Y50 Computer solution of Diophantine equations 11D25 Cubic and quartic Diophantine equations 11D41 Higher degree equations; Fermat’s equation 11P05 Waring’s problem and variants
heapsort
Full Text:
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