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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20
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1answer
379 views
+50

(Unsolved) In this infinite sequence, no term is a prime: prove/disprove.

$ 343,~ 34343, ~3434343, ~343434343,\ldots$ $\begin{array}\\ \color{Red}{343} &\color{Red}{: 7^3}\\ 34343 &: 61\times 563\\ \color{green}{3434343} &\color{green}{: 3\times 11^2\times ...
3
votes
1answer
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+500

Fermat's last theorem and $\mathbb{Z}[\xi]$

I heard that one can prove special cases of FLT by using unique factorization in $\mathbb{Z}[\xi]$ (whenever this is possible), where $\xi$ is a primitive $n$-th root of unity. How can one do this in ...
1
vote
1answer
76 views
+50

Lower Limit Prime Gap

The recent result of Zhang states $$ \lim \inf_{n\rightarrow \infty}(p_{n+1}-p_{n})<7\times 10^{7} $$ The upper bound is being optimized but if we assume $$ \lim \inf_{n\rightarrow ...
2
votes
1answer
88 views
+100

Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$

I'm trying to characterize the behavior of the the quantity: $$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$ subject to the constraints that $$ \sum \limits_{i = 1}^N ...