yspm's blog

By yspm, history, 2 months ago, In English

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.

Read more »

 
 
 
 
  • Vote: I like it
  • +24
  • Vote: I do not like it

By yspm, history, 3 months ago, In English

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.

Read more »

 
 
 
 
  • Vote: I like it
  • +24
  • Vote: I do not like it

By yspm, history, 7 months ago, In English

happyguy656 just became a Grandmaster and he wants you to help him to prove the below equation:

$$$\sum_{i=1}^{n} i\binom {n-1}{n-i}\begin{Bmatrix} {n-i}\\{k-1}\end{Bmatrix}=\begin{Bmatrix}n \\ k\end{Bmatrix}+(n-1)\begin{Bmatrix}{n-1}\\ k\end{Bmatrix}$$$

Let $$$\binom nm$$$ be $$$0$$$ when $$$n<m$$$ and n,k are given integers

Read more »

 
 
 
 
  • Vote: I like it
  • +47
  • Vote: I do not like it