id: 05694770
dt: j
an: 2010b.00392
au: Overholt, Marius
ti: Sums of two squares. (Summer av to kvadrat.)
so: Normat. 57, No. 3, 97-106 (2009).
py: 2009
pu: Nationellt Centrum för Matematikutbildning (NCM), Göteborgs Universitet,
Göteborg
la: NO
cc: F65
ut: proof of Fermat’s two squares theorem due to Heath-Brown; asymptotic
distribution; occurrence in prescribed intervals; analytic number
theory
ci:
li:
ab: 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 Heath-Brown 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.
rv: