[Math] “every prime number $p$ $(p>3)$ can be expressed sum of consecutive numbers ” is it true

Question

I'm finding some necessary and sufficient conditions for a integer $n$ to be a prime number. But I'm not sure if "every prime number $p \, ,(p>3)$ can be expressed sum of consecutive number" is true.
If it is right, I hope you help me prove that.

Thank you very much.

Answer 1

Every prime greater than or equal to $3$ is odd, and every odd is of the form $2k+1$ that is, $$\exists \,k \in \mathbb N \, : \,p=2k+1=\underbrace{(k)+(k+1)}_{\text{Sum of two consecutive integers}}$$