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 May 11th, 2011, 10:05 AM #1 Senior Member   Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0 Writing a prime as sum or difference of different primes Can any prime $p \ >3$ be written as $p= p_1 \pm p_2 \pm ... \pm p_n$, if $p_k$ are all primes different from p (not necessarily the k-th prime) and$p_i \neq p_j$ for all i,j? For example, $7= 5 + 2$ or $7= 17 - 5 - 3 - 2$ Let$p_n= p_{n-1} + k$ (in this case, I do mean the n-th prime). Then for the statement to be false, k must not be able to be written as $p_m= p_{m-1} + k$, otherwise$p_n= p_{n-1} + p_m - p_{m-1}$. So if there isn't an unique difference between two consecutive primes greater than 3, then the statement is true. Is it true that every even number greater than zero can be written as a difference of two or more pairs of consecutive primes?
May 11th, 2011, 11:29 AM   #2
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Re: Writing a prime as sum or difference of different primes

Quote:
 Originally Posted by proglote Can any prime $p \ >3$ be written as $p= p_1 \pm p_2 \pm ... \pm p_n$, if $p_k$ are all primes different from p (not necessarily the k-th prime) and$p_i \neq p_j$ for all i,j?
Almost surely, and you don't need to exclude 2 = 5 - 3 or 3 = 5 - 2.

Quote:
 Originally Posted by proglote Is it true that every even number greater than zero can be written as a difference of two or more pairs of consecutive primes?
Alphonse de Polignac conjectured that it would only take one pair. It hasn't been proven yet in the 160 years since.

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