sign in
english version rss feed

inria-00071634, version 1

Number of solutions to $(A^2+B^2=C^2+C)$ in a binade

Jean-Michel Muller 1, Jean-Louis Nicolas, Xavier-François Roblot

N° RR-4945 (2003)

Abstract: Let us denote by Q(N,[lambda]) the number of solutions of the diophantine equation $(A^2+B^2=C^2+C)$ satisfying N<=A<=B<=C<=[lambda]N-1/2. We prove that, for [lambda] fixed and N-->infinity, there exists a constant [alpha]([lambda]) such that Q(N,[lambda])=[alpha]([lambda])N+O_[lambda](N^7/8logN). When [lambda] =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)-->[root](a^2+b^2) in radix-2 floating-point arithmetic. By elementary arguments, Q(N,[lambda]) can be expressed in terms of four sums of the type S(u,v;f)=[SIGMA]_(u<=d<=v) ([SIGMA]_(1<=[lambda]<=f(d)) 1) where u and v are real numbers and f: [u,v] -->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 [lambda]=2, a precise upper bound for |Q(N,[lambda])-[alpha]([lambda])N| is given.

  • Domain : Computer Science/Other
  • Keywords : COMPUTER ARITHMETIC / CORRECT ROUNDING / DIOPHANTINE EQUATIONS
  • Internal note : RR-4945
 
  • inria-00071634, version 1
  • oai:hal.inria.fr:inria-00071634
  • From: 
  • Submitted on: Tuesday, 23 May 2006 18:22:19
  • Updated on: Wednesday, 31 May 2006 14:24:25
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...
all articles on CCSd database...