Problems & Puzzles:
Pythagorean triplets such that...
Sebastián Martín Ruiz sent the following nice
Pythagorean triplets of the form [k, n, p(n)]
found only two of them: [4, 3, 5] and [35, 12,
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
Found no more than the already known solutions, however the library
I used might have bugs. Searched up to [k, 1779586976727,
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