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 ...
Cave Johnson's user avatar
  • 5,397
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 $, $\...
Alexey Ustinov's user avatar
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 } ...
Jeremy Rouse's user avatar
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 = -\...
dan_fulea's user avatar
  • 1,796

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