# Questions tagged [modular-arithmetic]

Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $a-b$.

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### Is there a simple formula for $\binom{2n}{n} \pmod{n^3}$?

Is there a simple formula for the following? $$f(n) = \binom{2n}{n} \pmod{n^3}$$ I know $f(n) = 2$ iff $n$ is prime and greater than $3$, but I don't know anything about composite numbers.
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### There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of the form $6k+1$.

I observed that if $n$ is a composite number of the form $6k + 1$ then there are at least three divisors of $n - 1$ which do not divide $\phi(n)$ (Euler's totient function). Is this true in general? ...
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### I conjecture $3^{{2^n}}+1=2^1\prod_{k=1}^m p_k^1$ where $p_k$ are primes of the form $1+4k$, $k\in\mathbb{N}^*$

Let $T_n=3^{{2^n}}+1$, $n\geq 1$. I conjecture $$T_n=2^1\prod_{k=1}^m p_k^1$$ where $p_k$ are primes of the form $1+4k$, $k\in\mathbb{N}^*$. Note that the exponents on the $p_k$ are all $1$. ...
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### Could a nice principe, or at least a simpler alternative proof, be found regarding a lemma of Gauss

To prove the quadratic reciprocity law, Gauss needed the following lemma: If $p$ is a prime number congruent to 1 modulo 8, then there exists a prime $q<p$ such that $p$ is a non residue modulo $q$...
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### Proof that $a^2+b^3+c^4=k$ modulo $m$ always has a solution

Question Given integers $m\ge1$ and $0\le k<m$, I want to show that $a^2+b^3+c^4=k \pmod m$ always has a solution. Attempt It’s easy to show that it is enough to prove this when $m$ is a prime or a ...
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### Can long numbers be "3-palindromic"?

Question Let $n$ be a number with $d\ge9$ digits when written in number base $b\ge2$. Can $n$ be $3$-palindromic? That is, does there exist $b$, such that $n$ is simultaneously a ...
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### Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\to S$ satisfying $\sum \limits_{x\in T} x^{f(x)}\equiv a \pmod p$

This question is the third round of Iranian exam questions, which has not been answered for several years now. I think there are many people here, which may be able to solve this problem. From AOPS ...
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