Problems & Puzzles: Puzzles

 Puzzle 1052. Pythagorean triplets such that... Sebastián Martín Ruiz sent the following nice puzzle. Find Pythagorean triplets of the form [k, n, p(n)] I have found only two of them: [4, 3, 5] and [35, 12, 37] Q. Find more triplets or prove there are not any more?

During the week ending on Set 3, 2021, contributions came from Simon Cavegn and Oscar Volpatti

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Simon wrote:

Found no more than the already known solutions, however the library I used might have bugs. Searched up to [k, 1779586976727, 54443015485871]

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Oscar wrote:

I checked all primes below 10^14, finding no more solutions.
Actually, I only checked odd primes congruent to 1 mod 4.
The triplet [k,n,p] must be primitive, so it fits the parametric formula for suitable a and b:
p = a^2+b^2;
n = a^2-b^2 if n is odd;

n = 2*a*b if n is even (and divisible by 4).
But odd primes congruent to 3 mod 4 can't be expressed as the sum of two squares.

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