# Questions tagged [congruences]

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### Complexity of solving system of binary quadratic equations modulo $3$

A special case of this question and another question What is the complexity of solving system of binary quadratic equations modulo $3$? $f_i(x_i,x_j)=0 \bmod 3, \deg{f_i}=2$. Modulo $2$ can be ...
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### On the congruential equidistribution of $3^n \bmod 2^n$

I am interested in the asymptotic congruential equidistribution of $u_n = 3^n \bmod 2^n$ in the set of positive odd integers, as $n\rightarrow \infty$. Or the asymptotic congruential equidistribution ...
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### When are modular forms linearly independent modulo $p$?

Let $M_2(\Gamma_0(N))$ be the space of weight $2$ modular forms for $\Gamma_0(N)$ and let $f_1, \dots, f_r$ be a basis of normalized eigenforms for $M_2(\Gamma_0(N))$. Given a rational prime $p$, I'll ...
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### Modulo $x^2 + y^2 - 1$, is every homogeneous polynomial that is a square of a polynomial, necessarily of sum of squares of homogeneous polynomials?

I am hoping this question is alright for Math Overflow. I didn't get a definitive solution in Math Stack Exchange. Let $f(x, y) \in \mathbb{R}[x, y]$ be a homogeneous polynomial with real coefficients ...
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### A congruence for a product of binomial coefficients?

For every prime $p\geq 5$ one seems to have the congruence $$(-1)^{(p-1)/2}\prod_{k=0}^{p-1}{p-1\choose k}\equiv 1-p+\frac{3}{2}p^2-\frac{7}{6}p^3\pmod{p^4}\ .$$ (I have checked all primes up to $5000$...
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### Probability of satisfying the congruent mod equation

I'm wondering about the probability of picking three different numbers $x,y,z$ out of the set $[50]=\left\{ 1,2,3,...,50\right\}$ satisfying the equation: $$xyz\equiv \gcd(x,y,z)\mod 7$$ I started out ...
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### A conjecture involving $P_n=\prod_{k=1}^np_k$
For each positive integer $n$ let $P_n=\prod_{k=1}^n p_k$, where $p_k$ is the $k$th prime. Question. Is my following conjecture true? Conjecture. For any integer $n>1$, there are \$k,m\in\{1,\...