# Goldbach's conjecture

Created 23rd April, 2009 03:24 (UTC), last edited 23rd April, 2009 03:56 (UTC)

Nicholas Palevsky sent me a message asking which BeerCamp puzzle was about Goldbach's Conjecture — I guess he was thinking of the number shuffle problem as I haven't done one on Goldbach's Conjecture. But let's fix that now.

Goldbach's Conjecture is another problem to do with prime numbers. It states that every even number greater than or equal to four is the sum of two primes. For example:

• 4 = 2 + 2
• 6 = 3 + 3
• 8 = 3 + 5
• 10 = 5 + 5, 7 + 3

And so on. The conjecture hasn't been proved yet, but it seems quite a likely thing. The proofs for these sorts of simple number problems are way beyond my maths abilities, but they do have the nice property that they are quite easy to investigate with computer programs. A program won't lead to a proof, but it will bolster the claim of a conjecture.

Anyway, lots of people have been investigating this particular conjecture for some time, but in my quick research of the topic this morning I didn't find any mention of a variation which I will modestly refer to as Kirit's conjecture :)

Every even number can be written as the difference of two primes

Whether you consider zero to be even or not doesn't especially matter, the conjecture certainly holds for it anyway. For any positive number that you find a match for, reversing the two primes will give you the same negative number, i.e. 2 = 5 - 3 and -2 = 3 - 5.

Here are the first few* [*Each of these can be found multiple more ways too. E.g. 2 = 7 - 5, 13 - 11 and probably an infinity of other ways. It's enough to find any one of these.]:

• 2 = 5 - 3
• 4 = 11 - 7
• 6 = 13 - 7
• 8 = 11 - 3
• 10 = 17 - 7

What's the largest/smallest number that you can verify Kirit's Conjecture for?