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I hope this question is valid as I'm just curious. in a tweet from AlgebraFact I read the following:

"There are 17 ways to write 17 as a sum of primes": $17, 2+2+13, 3+3+11, 3+7+7, 5+5+7, 2+2+2+11, 2+3+5+7, 2+5+5+5, 2+2+3+3+7, 2+2+3+5+5, 3+3+3+3+5, 2+2+2+2+2+7, 2+2+2+3+3+5, 2+3+3+3+3+3, 2+2+2+2+2+2+5,2+2+2+2+3+3+3, 2+2+2+2+2+2+2+3$

It seemed to me a curious property and I would like to know if there are more primes that satisfy this property or if there is any result in this regard (another thing that seems strange to me is that 17 is included)

Luis Alexandher
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    I would expect the number of ways to decompose to grow more quickly than the prime itself; thus relatively few primes should behave the same way as 17. Perhaps you can find the counts for 5, 7, 11, 13 which should be small – Will Jagy Apr 29 '22 at 03:20
  • The fact that $17$ is included in the list is just because that makes $17$ ways. Whoever found this was playing and either would be fine. – Ross Millikan Apr 29 '22 at 04:08
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    It's tabulated at https://oeis.org/A000607/b000607.txt and it's pretty clear that $17$ is the only $n$ that's a sum of primes in $n$ ways. Much literature is cited at https://oeis.org/A000607 – Gerry Myerson Apr 29 '22 at 04:22
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    We can find more primes that have similar properties if we're more flexible about the properties we consider. For example, there are $35$ ways to write $35$ as a sum of odd primes; regrettably $35$ is not itself prime. – Misha Lavrov Apr 29 '22 at 05:35

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Here is an approach to prove that there are only finitely many numbers with this property. We only pay attention to expressions of the form $m \cdot 2 + n \cdot 3 +p\cdot 5 +7=N$. The point is that the number of expressions of this type grows faster than $N$, so there will eventually be far too many expressions of this type and we have ignored others. There are a lot of asymptotic approximations made, so you need to fill in a lot of blanks.

There are about $N/6$ ways to express $N$ as an unordered sum of $2$s and $3$s. If you only use $2$s there are $N/2$ of them, then you can replace $3\ 2s$ by $2\ 3s$ up to $N/6$ times. For each of these $N/6$ expressions, we can replace $2+3$ by $5$, or $5\ 2$s by $2\ 5s$, or $5\ 3$s by $3\ 5$s. For each sum of $2$s and $3$s there are at least $N/15$ replacements we can do of this sort. We therefore have at least $N^2/90$ ways to express $N$ as a sum of $2,3,5$s For a given $q$ we have at least $\frac {(q-7)^2}{90}$ ways to express $q$ as a sum of primes, which exceeds $q$ when $q \gt 100$. There are not many numbers to check. In fact the number of expressions will grow much faster than this, so once it is greater than $N$ you will be done. $17$ should be the only one.

Ross Millikan
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