Problems & Puzzles: Puzzles
Puzzle 162. P2 + Q2 = p2 +q2
Jean Claude Rosa poses the following puzzle:
The first four solutions found by J. C. Rosa are:
1. Can you find six more solutions?
Relaxing the primality condition he also has found some curios patterns:
(numbers in bold letter are prime numbers)
2. Can you find a pattern like these shown above but using only odd numbers, none of which are ending in "5" neither are divided by "3" (satisfying these conditions it may happen that one equation member of the pattern could have prime all the four numbers )?
Jud McCranie sent the following 3 more solutions:
15697673 37784951 37679651 15948773
Bingo!... J.C. Rosa discovered that the third example gotten by Jud is the second member of one patter as the asked in the question 2:
(primes are in blue)
Does this pattern has another case with all the four members primes?
One day later, J.C. Rosa discovered that the first example found by Jud is also a member of another pattern:
Does this mean that all the prime-solutions are members of certain patterns?
Faride FiroozBakht found other interesting examples:
More solutions for Q.2 came from Jon Wharf (5/1/2003)
Jon Wharf also found more solutions to Q1:
He has explained to me his approach, and it's available to any reader on request.