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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. –  Feb 14 '20 at 07:57
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    I have checked the conjecture for primes $\leq1000$ without counterexamples. – YiFan Tey Feb 14 '20 at 08:07
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    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 '20 at 08:08
  • @Sil Yes, that is the conjecture. –  Feb 14 '20 at 08:12
  • This can probably be rephrased in terms of norms in quadratic integer rings, for instance representing $p = q^2 + 2$ can be rephrased as determining quadratic integers $q + \sqrt{-2} \in \Bbb Z[\sqrt{-2}]$ of Norm p; such questions are addressed in a book by David Cox on primes of the form $p = x^2 + my^2$. Maybe this is overkill though! – Edward Evans Feb 14 '20 at 08:13
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    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 Tey Feb 14 '20 at 08:22
  • @YiFan Thank you. I know about some of the related conjectures although this one I proposed seems to be entirely new. –  Feb 14 '20 at 08:24
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    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 '20 at 08:45
  • This is a weaker form of a conjecture I made that "Every positive integer $n>10$ can be written as a sum of a prime $p$ and two other primes (not necessarily distinct) $q$ and $r$ multiplied together." – TheSimpliFire Feb 14 '20 at 09:18
  • @TheSimpliFire It is not strictly weaker just because you did not state that p can be equal to either r or q. –  Feb 14 '20 at 09:23
  • Also posted to (but closed & deleted at) MO, https://mathoverflow.net/questions/352805/which-one-do-you-think-is-harder-to-settle – Gerry Myerson Feb 15 '20 at 21:49

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