10
votes
Accepted
A real-valued analogue of the Weierstrass $\wp$ Function
Sums of this kind are called Epstein Zeta function.
More generally, they are studied under the name of Lattice Sums. There is the book Lattice sums then and now devoted to this subject. There one can ...
9
votes
Counting points on elliptic curves
Corresponding sums of Legendre symbols are know as Jacobsthal sums $$J(u)=\sum_{x\mod p}\left(\frac {x^3+ux}p\right).$$
They are equal to $0$ for prime $p\equiv 3\pmod4$. If $p\equiv 1\pmod4
$, $\...
5
votes
Accepted
Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Regarding your first question, there are no solutions $v$ to your equation for the three values of $u$ given (at least assuming GRH). For $u = 97/72$ and $u = 103/78$, the $D^{2} = \text{ quartic in } ...
2
votes
Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Let us consider first the case
$$
\bbox[yellow]{\qquad u=\frac{97}{72}\ ,\qquad}
$$
and put the hands on the objects related to it. The quartic associated to this value is:
$$
(C)\ :\qquad
D^2 =
-\...
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