Greg Dresden, Kylie Hess, Saimon Islam, Jeremy Rouse,
Aaron Schmitt, Emily Stamm, Terrin Warren and Pan Yue
Vol. 12 (2019), No. 4, 585–605
DOI: 10.2140/involve.2019.12.585
Abstract
Using Fermat’s two squares theorem and properties of cyclotomic
polynomials, we prove assertions about when numbers of the form
can be expressed as the sum of two integer squares. We prove that
is the sum of two
squares for all
if and
only if
is a square.
We also prove that if
,
is odd, and
is the sum of two
squares, then
is the sum
of two squares for all
,
.
Using Aurifeuillian factorization, we show that if
is a prime and
, then there are either zero
or infinitely many odd
such that
is the sum of
two squares. When
, we
define
to be the least
positive integer such that
is the sum of two squares, and prove that if
is the sum of
two squares for
odd, then
,
and both
and
are sums of two squares.
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