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 28 Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24}$ 13 If $x^4 \equiv -1 \mod p$ then $p \equiv 1 \mod 8$ 10 Find $\lim\limits_{n\to+\infty}\sqrt[n]{\prod\limits_{k=1}^{n}{n\choose k}}$ 10 How to find $f$ if $f(f(x))=\frac{x+1}{x+2}$ 10 $x^4 - y^4 = 2z^2$ has no solution

### Reputation (5,050)

 +20 Question in finding a new $\mathbb{Q}$-basis for $F/\mathbb{Q}$. +45 Show $\ln\left(\frac{n+(-1)^{n}\sqrt{n}+a}{n+(-1)^{n}\sqrt{n}+b} \right)=\frac{a-b}{n}+\mathcal{O}\left(\frac{1}{n^2} \right)$ +65 Two randomly chosen coprime integers +5 Converse of Chinese Remainder Theorem

### Questions (2)

 10 Converse of Chinese Remainder Theorem 1 $p$-adic $n$-th root of unity and $\exp(2\pi i /n)$

### Tags (131)

 76 number-theory × 26 28 elementary-number-theory × 12 66 sequences-and-series × 18 28 closed-form 52 calculus × 10 27 linear-algebra × 14 39 integration × 6 27 limits × 8 36 definite-integrals × 3 26 matrices × 13

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