Problems & Puzzles: Puzzles

 Puzzle 676 primes in a row such that... Bernardo Recaman Santos and Carlos Rivera, as a result of a short & swift email interchange, composed the following puzzle: Q1. Find the smallest prime P (greater than 11) such that it is possible to split and arrange all the primes,  2, 3, 5, 7, 11, 13, ..., P, in two rows in each of which the difference between adjacent primes is always a perfect square. ( A single row is not possible because primes of the form 4n+1 cannot mix with primes of the form 4n+3.).   Q2. Find the smallest prime P  such that it is possible to split and arrange all the primes,  2, 3, 5, 7, ..., P, in two circles (or necklaces) in each of which the difference between adjacent primes is always a perfect square.

Contributions came from Giovanni Resta, Jean Brette, Hakan Summakoglu & Emmanuel Vantieghem.

***

All of them found the same solutions for both questions, except Jean Brette who only found solution for Q1 "working at hand".

The smallest partition in two rows is

(5, 41, 37, 53, 17, 13, 29), and
(31, 47, 11, 2, 3, 19, 23, 7, 43).

The smallest range of primes that can partitioned into two cycles
is 2,3,...,353 (71 primes):

Two examples of these two cycles are:

{5, 41, 37, 101, 137, 281, 181, 197, 97, 61, 317, 313, 349, 353, 29,
173, 157, 257, 241, 277, 293, 229, 193, 337, 13, 17, 53, 89, 233,
269, 73, 109, 113, 149, 5}

and

{2, 3, 103, 139, 283, 347, 331, 7, 11, 47, 31, 131, 167, 151,
251, 107, 71, 67, 211, 311, 307, 271, 127, 191, 227, 263, 199, 163,
179, 79, 223, 239, 43, 59, 23, 19, 83, 2}

or:

{5,41,37,101,137,281,181,197,97,241,277,293,229,193,337,353,349,
313,317,61,257,157,173,29,13,17,53,89,233,269,73,109,113,149,5}

and

{2,11,47,191,227,211,311,307,271,127,131,31,67,71,107,251,151,
7,331,347,283,139,103,167,23,59,43,239,223,79,179,163,263,199,
3,19,83,2}

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