Problems & Puzzles: Puzzles

Puzzle 528. Prime quadratic residues JM Bergot sent the following puzzle: Take any odd prime P and its neighbors P-1 and P+1. Find the prime quadratic residues of P and put them into Set Q. Create the Set G from all the primes found in the Goldbach decompositions of both P-1 and P+1. Will Set Q be a subset of Set G? Example: For 29, the prime quadratic residues are 5, 7, 13, and 23. From 28 and 30, one finds 5+23 (plus 11+17) and 7+23, 13+17 (plus 11+19)  to give all the prime quadratic residues of 29. So Q is a subset of G.

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Contributions came from Emmanuel Vantieghem and Jan van Delden

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Emmanuel wrote:

First, I assume we should restrict Q to be the set of ODD quadratic residues mod p, since 2 is never a 'Goldbach prime' for p-1 or p+1.
The question then has a negative answer.
The first counterexample I found was 37 : 3 is quadratic residue, but 36-3 and 38-3 are composite.
Nevertheless, I found it interesting to make a list of all the primes for which Q is a subset of G (and which I would like to call 'Bergot Primes'). The only members I could find are :
2,3,5,7,11,13,17,19,23,29,41,47,53,89,113
If there is an additional prime, it must be greater than the 5000-th.
The primes for which the intersection of Q and G is non empty are much numerous. Here are the first ones :
11,13,17,19,23,29,41,43,47,53,61,71,73,79,83,89,101,103,107,109,113,131,137,167,181,193,197,199,227,229,233,241,269,281,283,293
I found 383 such primes among the first thousand primes. There seem to be a lot of primes such that the intersection of Q and G is empty.

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Jan wrote:

Hardly. The only primes (in a small search, the first 1000 primes) that qualify are: 2,3,5,11,13,19,29 and 53; of which the first 3 are trivial since the set S is empty in these cases.

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