Problems & Puzzles: Puzzles

Problem 81.  Pythagorean triangles with triangular sides

In the sequence A213188 we may read the following comment:

"It is possible for both of the legs to be triangular numbers as well as the hypotenuse. The only known example is 8778^2 + 10296^2 = 13530^2. - Andrew Howroyd, Aug 17 2018"

Q. Can you try to get one more solution?


During the week 12-19 Feb 2021, contributions came from Simon Cavegn, Paul Cleary


Simon wrote:

Found no more than the known solution.
Checked range: Added first 470'000 squared triangle numbers to first 10'000'000 squared triangle numbers, and could not find another solution.
First create an Array and a HashSet of the first 10'000'000 squared triangle numbers.
Then multithreaded loops to sum up each two of them: c_square = a_square + b_square.
Then check all the conditions c_square has to fulfill. The order of the checks matters to optimize speed.
First check is a very fast way to rule out most of cases when c_square cannot be a square number:
if ((0x202021202030213L & (1L << (int)(csquare & 63))) > 0)
The constant encodes all possible ways of the last 6 bits of square numbers.
Then do slower checks e.g. if c_square is in the HashSet of triangle numbers, and if c_square is a square number.


Paul wrote:

I wasn't able to find another solution where all 3 sides of the triangle were triangular numbers, I searched for the short leg of the triangle <= 1354895857440.


I did however find all solutions where the short leg and one of the two other sides were triangular.  See attached text file.


The format shows if its a leg or the hypotenuse that is triangular.  The single number is the triangular short leg and the pairs of numbers are the other leg and hypotenuse.


Example in the case of these numbers


Hyp , 91 , {{312,325},{588,595}} shows the short leg and hypotenuse are triangular numbers of the triangles.


{91, 312, 325} and

{91, 588, 595}.



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