All Questions
10 questions
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
12
votes
0
answers
704
views
Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?
Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...
- 12.6k
6
votes
1
answer
752
views
Question on the Sato-Tate conjecture
Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. For a good prime $p$, define $\theta_{E}(p)$ by
$$\cos\theta_{E}(p)=\frac{p+1-N_{p}(E)}{2\sqrt{p}}\quad (0\leq \theta_{E}(p)\leq \pi).$$
I ...
- 63
6
votes
0
answers
98
views
Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.
Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform.
Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...
- 221
4
votes
1
answer
515
views
Recent works on the Hardy-Littlewood conjecture on primes represented by quadratic polynomials
I have been working on my master's thesis which is about the equivalence of the Hardy-Littlewood conjecture on primes represented by quadratic polynomials and the Lang-Trotter conjecture for CM ...
- 309
3
votes
0
answers
330
views
Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?
Let $g(n)$ be the Dirichlet inverse of the Euler totient function:
$$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
and let $f(x,y)$ be the elliptic equation:
$$f(x,y)=x^3 - x^2 - y^2 - y$$
Show that the ...
- 1,183
2
votes
1
answer
347
views
Which composites pass this probabilistic primality test?
If a composite integer resembles a prime too closely, it must pass
algorithmic tests designed to find primes and in addition avoid nontrivial
factorization.
Given an integer $p$, assume it is prime ...
- 25.4k
2
votes
0
answers
110
views
+50
How to apply Pohlig Hellman using a very limited set of auxiliary inputs in that case?
So I was reading about Talotti, Paier, and Miculan - ECC’s Achilles’ Heel: Unveiling Weak Keys in Standardized Curves. The underlying idea is to lift the discrete logarithm problem to $\mathrm{prime}−...
- 87
2
votes
0
answers
147
views
Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 1,422
2
votes
0
answers
124
views
On the elliptic curve $X^3+6d^2X-7d^3 = Y^2$ and the ellipse $p^2+3q^2-d = 0$?
From the ellipse $p^2+3q^2 - d = 0$ we can find a solution to the equation,
$$a^3+b^3+c^3 = (c+m)^3$$
if we solve the elliptic curve,
$$E:=X^3+6d^2X-7d^3 = Y^2$$
More details can be found in this MSE ...
- 12.6k