# Is this elementary-number-theoretic conjecture already mentioned somewhere or it is entirely new?

While I was doing some research in elementary number theory I discovered some regularities that seem to be very promising.

First, function $$d$$ can be defined as $$d(n)=d(\prod_{i=1}^{k(n)} p_i^{r_i})=\sum_{i=1}^{k(n)}r_i$$ where $$n= \prod_{i=1}^{k(n)} p_i^{r_i}$$ is unique prime factorization of $$n$$ and it can be said that natural number $$n>1$$ is of degree $$w$$ if and only if $$d(n)=w$$.

So, for example, prime numbers are natural numbers of first degree.

The conjecture I would like to propose is the following:

Every prime number $$\geq7$$ is the sum of a prime number and number of second degree.

As noted in the comments, the conjecture can be rephrased very simply as:

Every prime number $$p\geq 7$$ can be written as $$p=qr+s$$ where $$q,r,s$$ are primes.

• @Rahul Only relatively small ones. – user750262 Feb 14 at 7:57
• I have checked the conjecture for primes $\leq1000$ without counterexamples. – YiFan Feb 14 at 8:07
• So the conjecture is: Every prime $p\geq 7$ can be written as $p=q+r^2$ or $p=q+rs$ where $q,r,s$ are (not necessarily distinct) primes. If so, I did just a quick check up to $p \leq 2\cdot 10^6$ and found no counterexample. – Sil Feb 14 at 8:08
• Related conjectures are Lemoine's Conjecture which says that every odd $n>5$ is of the form $n=p+2q$ for primes $p,q$, and the Sum of prime and semiprime conjecture. You might also want to look at Chen's Theorem. – YiFan Feb 14 at 8:22
• All of $q,r,s$ cannot be odd since then $p$ would be even. So one of them is equal $2$, so either $p=2+qs$ or $p=q+2r$. – Sil Feb 14 at 8:45