# Questions tagged [congruences]

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### Two conjectures for primes $p\equiv 1\pmod 8$

Motivated by my paper Quadratic residues and quartic residues modulo primes [Int. J. Number Theory 16 (2020), 1833-1858], here I pose two new conjectures for primes $p\equiv1\pmod8$ based on my ...
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### Conjectural congruences for numbers related to Littlewood-Richardson coefficients

For $n \geq 0$, let $a_n$ be the square of the Euclidean length of the vector of Littlewood-Richardson coefficients of $\sum_{\lambda \vdash n} s_\lambda^2$, where $s_\lambda$ are the symmetric Schur ...
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### System of congruences with bound condition

Let $m$ be a positive integer divisible by $6$, and let $q$ be one of $8,9$, or a prime $\gt 3$. Question : Is there always an $x\in [1,m]$, coprime to $m$, such that $x\not\equiv\pm 1 \ \mod{q}$ ? ...
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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 $=\left\{ 1,2,3,...,50\right\}$ satisfying the equation: $$xyz\equiv \gcd(x,y,z)\mod 7$$ I started out ...
1 vote
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### Corollaries of the halo conjecture that do not involve the eigencurve

In the theory of p-adic modular forms there is a certain construction called the Coleman-Mazur eigencurve. The spectral halo conjecture roughly states that if you remove a closed subdisc of the weight ... 156 views

### Congruences of modular forms modulo other modular forms

Congruences between modular forms are certainly a big topic in number theory, maybe with $$E_{p-1}\equiv 1 \mod p \qquad \text{for a prime }p\geq 5$$ as the easiest example. Sometimes, $p$ might be ...
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### $\varphi(m+n)\mid n$ for some positive integer $n$

Let $\varphi$ be Euler's totient function. If $p$ is a prime, then $\varphi(1+n)=n$ for $n=p-1$. Question. Is it true that for each integer $m>1$ there is a positive integer $n\le m^2-m$ such that ...
1 vote
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1 vote
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### Efficient scissors congruence between efficiently describable convex polytopes and simplex?

Is there a convex polytope in $\mathbb R^n$ describable by only $O(poly(\log n))$ half-plane inequalities with positive volume (so at least $n+1$ vertices) such that the standard simplex has a ...
### Reversing the CRT: Is $5$ tough?
Given odd primes $p\ne q$, by the CRT we can find an integer $x$ such that $x\equiv 2^{p-1}\pmod q$ and $x\equiv 2^{q-1}\pmod p$. Can this procedure be reversed? For which integers $x$ there exist ...