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Topic: A conjecture for sets of four primes
Replies: 12   Last Post: Oct 14, 2013 8:11 PM

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quasi

Posts: 10,305
Registered: 7/15/05
Re: A conjecture for sets of four primes
Posted: Oct 12, 2013 3:40 PM
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Here is a revision of Ludovicus' conjecture, edited to
eliminate early counterexamples:

=========================================================

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

=========================================================

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 4-tuples
of odd primes whose sum is n. So many 4-tuples 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 4-tuples of odd
positive integers whose sum is n, and choose them randomly,
one at a time, until one of those 4-tuples (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 4-tuples whose sum is n easily exceeds the expected
required number of random odd 4-tuples whose sum is n such
that one of those odd 4-tuples satisfies the sum of squares
condition.

quasi



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