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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8
votes
0answers
140 views
+50

Maximum integer not in $\{ ax+by : \gcd(a,b) = 1 \land a,b \ge 0 \}$

Ryan asked about a variation of the coin problem, which was whether for any coprime natural numbers $x,y$ every sufficiently large natural number is $ax+by$ for some coprime natural numbers $a,b$. ...
9
votes
1answer
110 views
+150

Every non-increasing sequence of polynomial towers stabilizes — Finitary proof

In this question we are concerned only with positive integers $\mathbb N$ and other finitary objects that can be encoded using integers. A term function means a total computable function $\mathbb ...
35
votes
3answers
585 views
+200

Prove that there exist infinitely many integers $(n^{2015}+1)\mid n!$

I conjecture that there exist infinitely many integers $n$ such that $$(n^{2015}+1)\mid n!.$$ I have seen a simpler problem that there exist infinitely many integers $n$ such that $(n^2+1)\mid ...
2
votes
0answers
87 views
+50

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
1
vote
0answers
54 views
+50

Special Case of Composite mersenne number mod p

We want to investigate if a composite mersenne number $p|2^{qb}-1$ where $p\nmid2^q-1$ ,$q,p$ are primes, $p=1+6qb,\ qb\equiv1(mod64) $ and $b$ is an odd number. In general for $$\begin{align*} ...
2
votes
0answers
96 views
+100

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...