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Who will be the first to prove that (with one exception) the sum of any Prime Pair is always divisible by twelve?
The first prime pair I happened to think of was 29 and 31Originally posted by Tyger
Who will be the first to prove that (with one exception) the sum of any Prime Pair is always divisible by twelve?
Originally posted by Tyger
Perhaps I should come up with some more teasers.
A couple of years ago I had some very bad flu, not debilitating but hung on for a long time and I was looking for a puzzle to make sure my brain was still working. I fiddled around and found that if a number could be represented as the sum of two squares its factors could also be represented that way. Checked it up to n=1,000 and decided to prove it. Wasn't very difficult, only had to think about it three times. My usual way to solve problems involves working with pencil and paper a little till some kind of progress is made then going on to other things. I usualy wake up at about three in the morning with another part worked out in my head, think about it a little more, maybe hit pencil and paper in the daytime, and a couple of nights later find another part. So I largely do it at a subconsious level and it involves putting the problem in my imagination, and the answer may arrive at the oddest times.
BTW the sum of squares problem was solved long ago by Euler and Fermat, but my proof involved complex notation and made it easier to see how many different ways a number could be so represented.
3 and 6 are not relatively primeOriginally posted by Hurkyl
Hrm, I must be misunderstanding the puzzle... consider 45:
45 = 3^2 + 6^2
But 3 is a factor of 45, and 3 is not the sum of two squares.