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 13 Limit in number theory 9 Find all integers $x, y, u, v$ for which holds: $x^2 + y^2 = 3(u^2 + v^2)$. 8 Prove $13^{17} \ne x^2 + y^5$ 7 Find $\lim_{n \to \infty} \left(n - \sum_{k=1} ^{n} \cos \frac{\sqrt{k}}{n} \right)$ 6 Could prime numbers be defined like this?

### Reputation (2,359)

 +10 How to estimate $\sum_{p\leqslant x}\sum_{q\leqslant x}\frac{1}{p+q}$? +25 Comparing two rational approximation of the same minimum +35 Lower bound for diagonal Ramsey numbers +10 Show that $(\binom{p^2}{p} -p )$ is divisible by $p^5$, for every prime number $p, p\ge 5$

### Questions (19)

 9 A006517: Numbers with $n\mid 2^n+2$ 8 If $\tan(x_1) \cdots\tan(x_n)=1$ for acute $x_i$, then does it follow that $\cos(x_1)+\cdots+\cos(x_n) \leq n\sqrt{2}/2$? 8 Why does $\int_0^{2\pi} (1+2\cos(x))/(5+4\cos(x))\,dx$ vanish? 5 Counting primes $p\equiv 1\pmod 4$ 4 Low-degree polynomial $T\in\mathbb F[x,y]$ with $T(P(z),Q(z))=0$

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 60 number-theory × 19 19 prime-numbers × 7 44 elementary-number-theory × 14 18 group-theory × 9 25 diophantine-equations × 10 14 calculus × 4 20 modular-arithmetic × 6 11 combinatorics × 6 20 limits × 2 10 real-analysis × 3

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