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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.

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  • $\begingroup$ @Rahul Only relatively small ones. $\endgroup$ – user750262 Feb 14 at 7:57
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    $\begingroup$ I have checked the conjecture for primes $\leq1000$ without counterexamples. $\endgroup$ – YiFan Feb 14 at 8:07
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    $\begingroup$ 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. $\endgroup$ – Sil Feb 14 at 8:08
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    $\begingroup$ 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. $\endgroup$ – YiFan Feb 14 at 8:22
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    $\begingroup$ 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 $p=q+2r$. $\endgroup$ – Sil Feb 14 at 8:45

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