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Re: Twin prime conjecture restated without reference to primes
Posted:
Sep 1, 2003 6:54 PM


uprho@yahoo.com (Paul A. Tanner III) wrote in message news:<8c28b591.0309010825.1b4adb4d@posting.google.com>... > In the midnineties, I stumbled across a trivial way of restating the > twin prime conjecture without reference to primes. I've sat on it > since it's so trivial, but found that others thought it a fun bit of > trivia when I shared it. Perhaps the following has been shared many > times before, but I've not seen it elsewhere. So, thinking that some > out there might find it an enjoyable bit of trivia on the side, here > it is: > > To eliminate all positive c such that 6c1 or 6c+1 is a composite, > take the cross product of the set of all positive integers of the form > (6a+1) with itself: (6a+1)(6b+1) = 6(6ab+a+b)+1 = 6c+1 for all > positive a,b and for all 4 combinations of "+" and ""(only 3 are > necessary, obviously). > > So the twin prime conjecture is true if and only if C = {c  6ab+a+b > = c for some a,b} leaves infinitely many positive integers uncovered. > > For economy, let C = {6ab+a+b), and so let R = {6abab}, S = S_1 = > {6aba+b} = S_2 = {6ab+ab}, and T = {6ab+a+b}. Although S = S_1 = > S_2, I note both because of the matrical representations below, where > S_1 and S_2 are the transposes of each other: > > C forms an infinite symmetric matrix M (that is actually just a > modified subset of the "infinite multiplication table" on the positive > integers), where each row or column is an ordered subset of the > residue classes +a modulo 6a1 or modulo 6a+1: > > 4 6 9 11 14 16 . . . > > 6 8 13 15 20 22 . . . > > 9 13 20 24 31 35 . . . > > 11 15 24 28 37 41 . . . > > 14 20 31 37 48 54 . . . > > 16 22 35 41 54 60 . . . > > . . . . . . > . . . . . . > . . . . . . > > So the twin prime conjecture is true if and only if M leaves > infinitely many positive integers uncovered. > >
Hindsight really is 20/20, is it not? Although the context makes it clear for some, I should have started my original post with:
"All variables are positive integers."
Anyway, below is a little extra bit of trivia that I stumbled across also in the midnineties along with the material in my original post, since it's similar to that material. Some have found it a bit of fun on the side, and I hope that you do, too:
Theorem.
Where a and b are positive integers with b > a:
1) Given prime p = 6a1 and composite q = 6b1: p  q if and only if p  ba 2) Given prime p = 6a1 and composite q = 6b+1: p  q if and only if p  b+a 3) Given prime p = 6a+1 and composite q = 6b1: p  q if and only if p  b+a 4) Given prime p = 6a+1 and composite q = 6b+1: p  q if and only if p  ba
Proof.
Where n is some positive integer:
1) [p  q] <=> [(6a1)(6n+1) = (6b1)] <=> [6(6ann+a)1 = 6b1] <=> [6ann+a = b] <=> [(6a1)n = ba] <=> [p  ba] 2) [p  q] <=> [(6a1)(6n1) = (6b+1)] <=> [6(6anna)+1 = 6b+1] <=> [6anna = b] <=> [(6a1)n = b+a] <=> [p  b+a] 3) [p  q] <=> [(6a+1)(6n1) = (6b1)] <=> [6(6an+na)1 = 6b1] <=> [6an+na = b] <=> [(6a+1)n = b+a] <=> [p  b+a] 4) [p  q] <=> [(6a+1)(6n+1) = (6b+1)] <=> [6(6an+n+a)+1 = 6b+1] <=> [6an+n+a = b] <=> [(6a+1)n = ba] <=> [p  ba]
Paul



