### yspm's blog

By yspm, history, 2 months ago, I've heard that if a number $n$ has quadratic surplus under a odd prime $p$ so it has around 50% accuracy to be a perfect square number.

Is it correct? How to prove?

If so,perhaps there exists a method to judge whether a big number $n$ is a perfect square number or not is that random amount of odd primes which are not divisors of $n$ and find Quadratic surplus of $n$ mod $p$. Failure appears means $n$ isn't a perfect square number.

By yspm, history, 3 months ago, I met a problem recently which need to prove the below conclusion.

$\exists B,\forall n\ge B,\exists p>\sqrt n, \text{p is prime,(2p+1) is prime}$

There is a example that $n=13,p=5,2p+1=11$ meet this constraint

I wrote something and check it's correct when $40\le n\le 10^5$ but is there any complete proof for such B exists?

I've learnt Bertrand-Chebyshev Theorem that there exists a prime $p\in[n,2n]$ and it may help.

Thank you.

By yspm, history, 7 months ago, $\sum_{i=1}^{n} i\binom {n-1}{n-i}\begin{Bmatrix} {n-i}\\{k-1}\end{Bmatrix}=\begin{Bmatrix}n \\ k\end{Bmatrix}+(n-1)\begin{Bmatrix}{n-1}\\ k\end{Bmatrix}$
Let $\binom nm$ be $0$ when $n<m$ and n,k are given integers