quasi
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Re: A conjecture for sets of four primes
Posted:
Oct 12, 2013 3:40 PM


Here is a revision of Ludovicus' conjecture, edited to eliminate early counterexamples:
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If n = k^2 where k is an even positive integer with k > 6, then there exist odd primes p,q,r,s such that
(1) n = p + q + r + s
(2) p^2 + q^2 + r^2 + s^2 is a perfect square
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Some comments.
The conjecture is probably true.
But it has nothing to do with n being square  nothing at all.
It has almost nothing to do with p,q,r,s being prime. The only tie to primes is that, by the Goldbach conjecture (or the Ternary Goldbach Theorem) there are _lots_ of distinct 4tuples of odd primes whose sum is n. So many 4tuples that, in general, one of them is sure to satisfy the sum of squares condition.
To understand this point, consider the following question ...
Take any large even positive integer n (with say, n > 10^4, just to be safe). Consider all possible 4tuples of odd positive integers whose sum is n, and choose them randomly, one at a time, until one of those 4tuples (a,b,c,d) is such that a^2+b^2+c^2+d^2 is a perfect square. As a function of n, what is the expected number of choices required?
My point is that, for sufficiently large even n, the number of odd prime 4tuples whose sum is n easily exceeds the expected required number of random odd 4tuples whose sum is n such that one of those odd 4tuples satisfies the sum of squares condition.
quasi

