All Questions
13 questions with no upvoted or accepted answers
9
votes
0
answers
324
views
Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
- 195
5
votes
0
answers
160
views
Reducibility of $f(x)^{2^n}+1$ and $f(x)^{2^n}+g(x)^{2^n}$
Related to generalized Fermat numbers.
Let $f(x),g(x)$ be coprime polynomials with integer coefficients.
Assume that if $f(x)$ or $g(x)$ are of the form $h(x)^k$ then $k$ is power
of two.
Q1 Is it ...
- 25.4k
5
votes
0
answers
370
views
Large prime factors of n²+1
Iwaniec proved (and many people extended) that the number of $n \le x$ for which $n^2+1=P_2$ (product of at most two primes) is $\gg x/\log x$. I am wondering what is known/can be proved for the ...
- 51
5
votes
0
answers
205
views
Is there a polynomial version of Wilson's theorem which can avoid Cramer flavored conjectures?
Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.
Is there a version ...
- 13.9k
4
votes
0
answers
187
views
Small solutions of $f(x_1,...,x_n) \equiv 0 \pmod p$
Let $f(x_1,...,x_n)$ be polynomial with integer coefficients.
Is the following possible:
For almost all primes $p$ exist integers $X_1,...,X_n$
such that:
$f(X_1,...,X_n) \ne 0$
$f(X_1,...,X_n) \...
- 25.4k
4
votes
0
answers
182
views
Integer polynomial inducing a permutation of order $p$ on $\mathbb{Z}/p\mathbb{Z}$ for infinitely many $p$
Let $Q\in \mathbb{Z}[x]$ be a non-linear polynomial. Can there exist infinitely many primes $p$ such that $Q(\mathrm{mod}\:p)$ induces a permutation $\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$ ...
user158636
4
votes
0
answers
279
views
Analog of Euler's factoring technique
Is there an analog of Euler's Two Squares factoring theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Euler's two squares factoring states that numbers ...
- 13.9k
3
votes
0
answers
408
views
The second conjecture about the degrees of special polynomials
Define the congruence "modulo m" on exponential Taylor series following the previous post (A conjecture about the degrees of special polynomials)
It has been conjectured, that if we define the ...
- 237
2
votes
0
answers
121
views
How to choose a prime p s.t. n-th cyclotomic polynomial splits into as much as possible irreducible polynomials while p is almost constant size?
The reason I ask this question is that cyclotomic polynomial is critical to the construction of lattice-based cryptography. In most of the existing lattice-based cryptographic schemes, $n$ is usually ...
- 33
1
vote
0
answers
107
views
Polynomial divisible by unbounded primes with exponent one
Let $f(x)$ be squarefree polynomial with integer coefficients and
degree at least $3$.
Is it true that for all sufficiently large $n$, $f(n)$ is divisible
by prime $p$ with exponent one and $p$ is ...
- 25.4k
1
vote
0
answers
133
views
Primes which do not divide certain homogeneous polynomials
It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which ...
- 5,470
1
vote
0
answers
202
views
Prime generating polynomials
Continuation to this previous question.
According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
- 195
1
vote
0
answers
195
views
Lower bound on number of smooth values of polynomial at primes
Given a polynomial $f$, it is known believed that the number of smooth values of $f$ has a positive proportion (for fixed $u$, $\lim_{X\rightarrow\infty} \frac{|\{ n < X\ :\ f(n)\ is\ X^u\ smooth \}...
- 4,593