Problems & Puzzles: Puzzles

Puzzle 187. Triangles and Triangular numbers

A certain Russian man called 'Eduardo' sent the following two puzzling tasks to Leonid Durman who - in his turn - shared with me these questions. As a matter of fact both questions seems to be a kind of apart from the primes, but they are strongly charming, indeed, to avoid momentarily this requirement.

First two definitions:

A "Pythagorean Triangle" is a rectangular triangle (A, B, C) such that A^2 + B^2 = C^2, being A, B & C positive integers.

A number T is said to be a "Triangular Number" if you can find a k such that T = T(k) = k.(k+1)/2 (*)

The puzzling questions rescued by 'Eduardo' are:

Task 1:

Find other solutions than the given below, for a Pythagorean Triangle (A, B, C) such that A, B & C are triangular numbers (or argue that no more solutions exist)

The only known solution is A=8778, B=10296 & C=13530 (found by K. Zarankiewicz, according to L. Durman)

Task 2:

Find other solutions than the given below, for a Pythagorean Triangle (A, B, C) such that the perimeter P = A + B + C and the Area S = A*B/2 are triangular numbers (or argue that no more solutions exist)

The only known solution is A = 14091, B = 3312 & C = 14475 (found by 'Eduardo', according to L. Durman)

__________
* N is a triangular number
if 1+8N is a square.

Solution:

Yves Gallot found a second solution to Task 2:

A = 8013265, B = 3405996, C = 8707079

His algorithm is here:

- for S a triangular even number
- for P even such that P divides 2*S and P^2 <= 24*S
- check if P is a triangular number
- check if the triangle generated by P and S is Pythagorean

Only two solutions for task 2 of puzzle 187 exist for P <= 43150*43151/2 (P <= 930,000,000).

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