I was thinking about this and am wondering if it is true. Currently trying to look for a counter example, but haven't found anything yet.

Conjecture: $p^\alpha$ can be written as the sum of two primes, for any prime $p$, $\alpha \geq 2 \in \mathbb{N}$.

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    $\begingroup$ What about $121=11^2$? $\endgroup$ – abiessu Oct 11 '13 at 4:51
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    $\begingroup$ Primes (except 2) are always odd. $\endgroup$ – Ewan Delanoy Oct 11 '13 at 4:52
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    $\begingroup$ Or even $3^3$... $\endgroup$ – Noam D. Elkies Oct 11 '13 at 4:52
  • $\begingroup$ Ah looked at small squares first didn't consider 27... Wonder when this does hold true? i.e works for 2^2, 2^3, 2^4, 5^2, etc. $\endgroup$ – user95072 Oct 11 '13 at 4:58
  • $\begingroup$ Don't worry about it... $\endgroup$ – Noam D. Elkies Oct 11 '13 at 5:14

Let $p=3$ and $\alpha = 3$. Then $p^3=3^3=27.$ A contradiction would be, $$27=2+25=3+24=5+22=7+20=11+16=13+14=17+10=19+8=23+4.$$ There are no other possibilities, and so this conjecture is false.

The case of showing when this is true reminds me of Goldbach's Conjecture which states that "Every even integer greater than 2 can be expressed as the sum of two primes".

Here is my construction of the reason why there is no conjecture on the odds (or primes):

Let $p>2$ be a prime. Then $p$ is odd, hence $p^{\alpha}, \, \alpha \in \mathbb{N}$ is odd (product of odds is odd). Suppose there are two primes $a,b>2$ such that $a+b=p^{\alpha}$. Then one of $a,b$ is even while the other is odd (sum of evens is even, sum of odds is even). Hence one of $a,b$ is not prime. Therefore no power $\alpha$ of a prime $p>2$ can be expressed as the sum of two primes $a,b>2$.

I will include any proofs not provided on request.

Note that there are cases where a prime + 2 will produce a prime $p^1$

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    $\begingroup$ What about $5^2 = 23 + 2$? $\endgroup$ – user95072 Oct 11 '13 at 6:02
  • $\begingroup$ Note my edit. There are many examples like that. I think this lends itself to your earlier comment though, 2 gets to a prime $p^1$ and maybe other powers often. Nice ;) $\endgroup$ – J. W. Perry Oct 11 '13 at 6:05
  • $\begingroup$ @DennisMeng read: unless one of a,b is 2 $\endgroup$ – John Dvorak Oct 11 '13 at 6:05
  • $\begingroup$ @DennisMeng Primes greater than 2 are odd. I think I have that condition in there. Keep going though, I want a clean post. $\endgroup$ – J. W. Perry Oct 11 '13 at 6:05
  • $\begingroup$ Bah, didn't see the $a, b > 2$ part. Ignore me. $\endgroup$ – Dennis Meng Oct 11 '13 at 6:06

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