Nombre de solutions dans une binade de l'equation A^2+B^2=C^2+C

Abstract

(eng) Let us denote by Q(N,\la) the number of solutions of the diophantine equation A^2+B^2=C^2+C satisfying N\le A\le B\le C \le \la N-\frac 12. We prove that, for \la fixed and N\to \iy, there exists a constant \al(\la) such that Q(N,\la)=\al(\la)N+O_\la\left(N^{7/8}\log N\right). When \la=2, Q(2^{n-1},2) counts the number of solutions of A^2+B^2=C^2+C with the same number, n, of binary digits; these solutions are interesting in the problem of computing the function (a,b) \to \sqrt{a^2+b^2} in radix-2 floating-point arithmetic. By elementary arguments, (N,\la) can be expressed in terms of four sums of the type S(u,v;f)=\sum_{\substack{u\le d\le v\\ d \text{ odd}}} \left(\sum_{\substack{1\le A\le f(d) \\ 4A^2\equiv -1 \hspace{-1mm}\pmod{d}}} 1\right) where u and v are real numbers and f: [u,v]\longrightarrow \R is a function. These sums are estimated by a classical, but deep, method of number theory, using Fourier analysis and Kloosterman sums. This method is effective, and, in the case \la=2, a precise upper bound for |Q(N,\la)-\al(\la)N| is given.