
05694770
j
2010b.00392
Overholt, Marius
Sums of two squares. (Summer av to kvadrat.)
Normat. 57, No. 3, 97106 (2009).
2009
Nationellt Centrum f\"or Matematikutbildning (NCM), G\"oteborgs Universitet, G\"oteborg
NO
F65
proof of Fermat's two squares theorem due to HeathBrown
asymptotic distribution
occurrence in prescribed intervals
analytic number theory
Summary: The characterization of numbers representable as the sum of two squares in terms of their divisors has been known since the 17th century (Girard, Fermat) but not proved until 18th century (Euler). A refinement involving the number $R(n)$ of representations of $n$ as sum of two squares was proved by Jacobi using theta functions. This article starts out by reproducing a very elementary proof due to HeathBrown of Fermat's characterization of primes of type $4n+1$. Then it turns to discussing problems concerning asymptotic distribution, occurrence in prescribed intervals, infinite occurrences of patterns such as $n,n+h_1$, $n+h_2\dots h_k$ for fixed $h_i$ (It is shown that $n,n+1$, $n+2$ occurs infinitely often, but of course $n,n+1,n+2,n+3$ may never all be sums of two squares.) Those are compared with the corresponding statements for primes. One may note that the asymptotic behaviour of $B(x)=\sum_{n= a^2+b^2C\log x$ for some suitable constant $C$, but so far it has only been shown for $h>Cx^{\frac 14}$. But if we relax the condition to almost all such intervals, there is a complete solution.