Problems & Puzzles: Puzzles

Puzzle 871. Claudio Meller 1476

Claudio Meller, in his always interesting site posted on March 8, 2017, the entry 1476.

In this entry he shows a curio property of the pair if integers {184, 345}:

184^1+345^1 is a perfect square, 23^2
184^2+345^2 is a perfect square, 391^2
184^3+345^3 is a perfect square, 6877^2

He made two simple questions about this property:

żAre there other primitive pairs as 184 & 345? (it's hard to accept that there is only one primitive solution)
żAre there pairs {n1, n2} that produce perfect squares n1^e+n2^e for e=1,2,...m, for m>3?

The "primitive" condition stated above, avoid the non-primitive condition of the following infinite set of pairs-solutions: {184*z^2, 345*z^2} for z=2, 3, ...

Up today no one has produced an answer to both questions.

Q1. Would you try to respond both questions?

By my side, I worked a little bit in the generalization of the puzzle posted by Claudio.

My generalization is this one:

żAre there set of k integers {n1, n2, ... nk}, k=>2 such that n1^e+n2^e+...nk^e are perfect squares for e=1, 2, ...m, m=>3?

Here are just some of my results, for m=3:

  • For k=3, {n1, n2, n3}={108,124,129} and many more solutions

  • For k=4, {n1, n2, n3, n4}={2, 2, 22, 38} and many more solutions

  • For k=4, prime values = {29, 41, 709, 1721}; {977, 1033, 1117, 2957} and for sure many more.

  • But I was unable to find solutions for m>3 with any k=>2 value.

Q2. Are solutions for k=>2 and m>3 impossible?

Q3. For which k values are possible prime solutions?



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