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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12
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0answers
127 views
+50

How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
8
votes
0answers
157 views
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Solving (n+1)(n+2)…(n+k)−k = x^2

Let $n$ and $k$ be positive integers. Need to find all pairs of $(n,k)$ such that $$(n+1)(n+2) \cdots (n+k)−k = x^2,$$ where $x^2$ is a perfect square.
1
vote
1answer
130 views
+50

Why it is impossible for primitive Pythagoras triplets in integers to be all as powerful numbers?

I had seen an elementary proof for Fermat's last theorem at Quora. I had checked all the steps (around one page only),where I couldn't catch any error, but I was confused about the last step only ...
5
votes
1answer
76 views
+100

How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!