# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Game of replacing number with divisors

In a game , there are N numbers and 2 player(A and B) . ...
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### Which primes $p$ divide $q^q-1$ for a prime divisor $q$ of $p-1$

I am looking for (a formula) for all the primes $p$ less than or equal to $X$ with the following criteria: There is at least one prime $q$ dividing $p-1$ such that $p$ divides $q^q-1$. $7$, for ...
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### Subset of Coins with maximal value

Let $n \in \mathbb{N}$ with $n\ge 3$ be given. Assume that you have $k-1$ coins of value $1/k$ for all $k \in \lbrace 2,\ldots,n \rbrace$. Now you have to choose a subset of these given ...
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### What is the number of Sylow p subgroups in S_p?

I am reading the Wikipedia article entitiled Sylow theorems. This short segment of the article reads: Part of Wilson's theorem states that (p-1)! is congruent to -1 (mod p) for every prime p. One ...
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### Can this function be a new test for primality?

The following function returns always 0 only if a number is not prime. $$H(x)=\prod_{i=2}^{x-1}\left\{\left[\sum_{k=1}^{x/i}(-i)\right]+x\right\}$$ what do you think? Bye!
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### conjectured arithmetic properties of some continued fraction

Given the continued fraction found in this post and looking similar to the one in this post $$G(q)=\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q(1-q^2)^2}{1-q^5+\cfrac{q(1-q^3)^2}{1-q^7+\ddots}}}}$$ ...
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### Is the relationship between coprime integers and irreducible fractions biconditional?

Are coprime integers and irreducible fractions related biconditionally? That is, if two integers are coprime ($a$ and $b$ say) then the fractions $\frac{a}{b}$ and $\frac{b}{a}$ are both irreducible. ...
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### Maximum length of a string that has no substring divisible by a prime number $p$ is $p-1$?

What is the maximum length of a string of nonzero digits that has no substring that is divisible by a given prime number? I want to find a string of length n which has no substring divisible by the ...