### Answers (620)

 20 Prove that the sum of pythagorean triples is always even 14 A real function problem 12 Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$ 8 Finding a closed form for recurrence relations $a_n=na_{n-1}+1$ and $a_n=na_{n-1}+n$ 8 How to prove that $\lim\limits_{n\rightarrow \infty} \frac{1}{n^2}\sum\limits_{k=1}^{n}(n \bmod k)=1-\frac{\pi^2}{12}$?

### Reputation (15,520)

 +10 If $x^2+ax+b+1=0$ ($a,b\in\mathbb{Z}$) has integral roots, prove that $a^2+b^2$ is composite. +10 If $m$ and $n$ are relatively prime positive integers, prove that $m^{\phi(n)}+n^{\phi(m)}\equiv1\pmod{mn}$ +10 Proving complex series $1 + \cos\theta + \cos2\theta +… + \cos n\theta$ +15 How do i show that : for $x, y >0 ,\log (x+y)=\log{x}\cdot \log{y}$ has no integer solutions?

### Questions (11)

 5 Is there a way to show that $\sqrt{p_{n}} < n$? 4 What is the catch in this geometrical/number theory question? 2 Is $x-\sqrt{x}< p_{\pi(x)}\leq x$? 2 Prove that for any given sequence of digits, there is a perfect square starting with that sequence 2 Is this function really useless?

### Tags (220)

 225 sequences-and-series × 140 114 prime-numbers × 67 223 real-analysis × 115 112 limits × 57 192 number-theory × 104 87 inequality × 43 177 elementary-number-theory × 103 75 complex-analysis × 57 130 calculus × 63 66 combinatorics × 37

### Bookmarks (424)

 498 What are imaginary numbers? 473 “The Egg:” Bizarre behavior of the roots of a family of polynomials. 278 Help with a prime number spiral which turns 90 degrees at each prime 231 What is the importance of the Collatz conjecture? 221 Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

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