4 Solving the equation of the type $g\left( x \right) = \int\limits_0^x {f\left( t \right)dt}$ 2 Prove that no Fermat number is a $3$ rd power of an integer. 2 Proofs with prime numbers. 2 How many solutions for this equation using combinatorics? 1 Is there a closed form solution to $y \frac{dy}{dx} = \sqrt{x^2+y^2}+x$?

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### Questions (20)

 5 Prove that $\frac{1}{2} \lt \sum_{r=1}^{n} \frac{1}{n+r} \lt \frac{3}{4} , n>1$ [duplicate] 4 If $a_i\in\mathbb{R}$, $\omega^2+\omega+1=0$, and $\sum_{i=1}^n\frac{1}{a_i+\omega^k} =2\omega^{2k}$ for $k=1,2$, find $\sum_{i=1}^n\frac{1}{a_i+1}$. 4 Prove that, if $p$ is an odd prime number, then ${f(p)}=\binom{2p-1}{p-1}-1$ is divisible by $p^2$ 3 Finding a monic polynomial with integer coefficients having $\sqrt{2} + \sqrt{3} + \sqrt{5} + \sqrt{7}$ as one of its roots. 3 Prove that $\frac{\binom{2n}{n}}{n+1}$ is an integer. [closed]

### Tags (48)

 4 functions × 2 2 combinations × 3 4 rolles-theorem 2 algebra-precalculus × 3 3 elementary-number-theory × 8 2 conic-sections × 3 2 combinatorics × 6 2 solution-verification × 3 2 modular-arithmetic × 4 2 permutations × 2

### Bookmarks (16)

 146 Sorting of prime gaps 45 The product of $n$ consecutive integers is divisible by $n$ factorial 33 Probability that two random numbers are coprime is $\frac{6}{\pi^2}$ 21 Representing all rational numbers between $\dfrac{1}{2}$ and $1$ 13 How to prove there exist two elements in a positive integer sequence with bounded differences such that one is a multiple of the other?

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