@Jakobian If $P_n$ is true then $P_{n+1}$, this will prove that all $P_n$ are true for coubtably many $P$, but what if there are uncountable many of them?
I tried to use A) induction B) to prove that $\sum_{k} N(k)\ge m$ and $\sum_{k} N(k)\le m$ but I got stuck in both because I focused too much in the possibility that some $N(k)>1$
I was trying to generalise Fermat's theorem> If $p$ is a prime number and $p = 1(mod 4)$, then there exist positive integers $a$ and $b$ such that $a^2 + b^2 = p$.
Then I have a question: what is the least $n$ such that $\sum\limits_{k=1}^na_k^2 =p$ for all prime $p$, where $a_k \in \mathbb{N}\cup 0$
It appears from the numerical evidence that the answer to that is $n=4$, also it appears that not only primes can be generated that way but all integers.
It's very annoying to read a math book where, in a proof, the author refers to a theorem by its number—like "Theorem X.Y". Of course, I won't remember theorems by number, so I end up endlessly scrolling or flipping pages to find it. It would be much more helpful if authors named their theorems based on either the person who discovered them or what the theorem actually says.
