All Questions
6 questions
5
votes
1
answer
377
views
When $p(x)^2 \mid f(g(x))$?
Let $f(x),g(x),p(x)$ be non-constant polynomials with rational coefficients.
Is it true that for all $f$ exist $g,p$ such that $p(x)^2 \mid f(g(x))$?
Partial results:
$f(g(x))$ is divisible by square ...
- 25.4k
4
votes
2
answers
411
views
Find all possible rational values of a parametric quartic such that it is reducible
Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 +...
- 441
4
votes
1
answer
204
views
Groups suitable for algebraic group factorizations of integers
Quoting Wikipedia on Algebraic-group factorisation algorithm
Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose ...
- 25.4k
3
votes
1
answer
261
views
Could efficient solutions of $x^2+n y^2=A$ be related to integer factorization?
Let $n$ be positive integer with unknown factorization and $A$ integer with known
factorization.
According to pari/gp developers pari can efficiently find all solutions of:
$$x^2+n y^2=A \qquad (1)$$
...
- 25.4k
2
votes
1
answer
161
views
Fixed $a_p=p+1-\#E(\mathbb{F}_p)$ and $a_p \ne 0$ on an elliptic curve infinitely often for fixed curve over the rationals?
In this and this question we show that if $p=27a^2+27a+7$ is prime, then the order of the elliptic curve
$y^2=x^3+2$ modulo $p$ is either $p$ or $p+2$.
Q1 Can we unconditionally show that the order ...
- 25.4k
0
votes
1
answer
263
views
Find all possible rational values of the parameter of a parametric cubic such that it is reducible
Description: Given the following parametric cubic polynomials ${E}^{3}
- 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E
+ 135\, {\beta}_{\pm} \left({5\...
- 441