Jacobsthal sums, Legendre polynomials and binary quadratic forms
Abstract
Let
p>3
be a prime and
m,n∈Z
with
p∤mn
. Built on the work of Morton, in the paper we prove the uniform congruence:
&\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big) \equiv {-(-3m)^{\frac{p-1}4}
\sum_{k=0}^{p-1}\binom{-\frac 1{12}}k\binom{-\frac 5{12}}k
(\frac{4m^3+27n^2}{4m^3})^k\pmod p&\t{if $4\mid p-1$,}
\frac{2m}{9n}(\frac{-3m}p)(-3m)^{\frac{p+1}4} \sum_{k=0}^{p-1}\binom{-\frac
1{12}}k\binom{-\frac 5{12}}k (\frac{4m^3+27n^2}{4m^3})^k\pmod p&\text{if $4\mid
p-3$,}
where
(
a
p
)
is the Legendre symbol. We also establish many congruences for
x(modp)
, where
x
is given by
p=
x
2
+d
y
2
or
4p=
x
2
+d
y
2
, and pose some conjectures on supercongruences modulo
p
2
concerning binary quadratic forms.