# 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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### Find a positive integer such that half of it is a square,a third of it is a cube,and a fifth of it is a fifth power

I found this problem in Underwood Dudley book in Linear congruence and Chineese remainder theorem ,but I can't present it as system of Linear congruence Anyone have any idea? plz help me
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### Generalized Mersenne Prime Conjecture Combined $(a,b)$ pair

As read here, it is conjectured that for and pair of integers $(a,b)$ that are coprime, not perfect ${r^{\text{th}}}$ powers, and $-4ab$ is not a perfect ${4^{\text{th}}}$ power, then there should be ...
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### How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
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### How to formally use Taylor expansions for $n$th derivatives and generating functions?

When deriving Catalan numbers, the generating function takes on this form: $$C(x) = \frac{1}{2} (1 - \sqrt{1-4x}) = \frac{1}{2} (1 - f(x))$$ where $f(x) = \sqrt{1-4x}$ How does one formally show ...
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### Is $\sqrt2+\pi$ irrational?
From this, as a layman I wonder if the same goes for $\sqrt2+\pi$? How about $\pi+\log2$?