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 16 How prove this function $f(x)=x!-x^n$ is injective 12 How prove that $x_1 = x_{2000}$ implies $x_2 \ne x_{1999}$, where $x_{n+2}=\frac{x_{n}x_{n+1}+5x^4_{n}}{x_{n}-x_{n+1}}$? 12 How many positive integer solutions to $a^x+b^x+c^x=abc$? 11 About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$ 9 How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$?

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 +10 A polynomial has only real roots and all coefficients $\pm 1$. Prove the degree $<4$. +10 How many times can a $4^{th}$ degree polynomial be equal to a prime number? +10 Need help understanding proof to their being a prime gap $G(p_k, p_{k+1})$ with $p_k \leq n$ and $p_{k+1} - p_k > \frac{1}{2}\ln(n)$ +10 If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$

### Questions (2)

 5 Binomial asymptotic. 0 Polynomial/ Exponential diophantine equation

### Tags (78)

 77 inequality × 22 28 sequences-and-series × 6 59 number-theory × 19 27 analysis × 10 46 real-analysis × 14 27 diophantine-equations × 6 41 calculus × 14 23 polynomials × 9 39 elementary-number-theory × 10 21 complex-analysis × 6

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