The 666 EDP triples

 

Have a solution the equation

a² + b² = c²

Obviously, the answer depends on where we look for the solution. For example, there is no solution among the odd natural numbers.

In other cases, solutions are known or findable, but then questions arise: whether one or many solutions exist, and how they can be found.

Yes, in some cases it may be true that there is only one solution, for example if we are looking for it among single-digit natural numbers.

The situation is quite well known if solutions are among the natural numbers. Far from being the first the illustrious Pythagoras also dealt with this interesting and almost entertaining topic and therefore grateful posterity calls the solutions of the above equation consisting of natural numbers Pythagorean triples.

It must be agreed here that, in search of a solution, it is no necessery to find three numbers. To find the solution, it is enough to find any two of the three numbers (a, b, c). The third can be simply calculated from the two. This gives rise to various special task types in this topic, such as asking whether the square of a given "interesting" number can be written as the sum of two squares, or in how many ways.

But let's finally see what brought us together today!

The question is, is there a solution to the above equation in Egyptian fractions? (Egyptian fractions are the inverses of the natural numbers, the 1/n type numbers.) That is, we are interested in the solutions of the following equation:

1/a² + 1/b² = 1/c²

We immediately see that the "classic" Pythagorean triple, 3, 4, 5, does not offer a solution here in any configuration, and they do not lead to a result either individually.

We get a hard Diophantus equation. "At first glance" it can be guessed that there is a solution, but beyond a few connections, to which I will return, I could not find a solution formula, so in my impatience I left the sacred mathematics and went to the no less sacred programming. A little python program collected for me the solutions where “a” and “b” are less than 10,000 in minutes.

It just became 666. Interesting. I hope no one is superstitious.

Here is the claimant to the throne, which replaces Pythagoras' "classic" 3²+4²=5²:

1/15² + 1/20² = 1/12²

Another interesting thing is that among the 666 solutions, 42 pairs of solutions have different "b" values for the same "a" value. For example:

195, 260, 156

195, 468, 180

In the case of "b" (i.e., the larger value), there are already 50 pairs of solutions with the same "b" value, and a trio of solutions where the situation is the same:

2275, 7800, 2184

3250, 7800, 3000

5850, 7800 , 680

In the case of the "c" value, the situation is quite "dramatic". There are 164 pairs of solutions, 25 cases with three solutions, 15 cases with four solutions, one case with five solutions, two cases with six solutions and one case with seven solutions. This can also be said that the square of a certain Egyptian fraction (in the case 2520) can be set up in seven different ways as the sum of the squares of two Egyptian fractions:

2625, 9000, 2520

2664, 7770, 2520

2730, 6552, 2520

2856, 5355, 2520

2968, 4770, 2520

3150, 4200, 2520

3480, 3654, 2520

(I just note quietly: 2520 = 7x360. The shroud covering the secrets of Osiris is falling apart.)

Another point of interest: just at the border of the research area, python found two solutions where there are two adjacent numbers:

2020, 9999, 1980

7500, 10000, 6000

The examined solutions obviously cannot be called Pythagorean triples, I would rather suggest the name EDP triples, where the letters mean: Egyptian, Diophantine, Pythagorean.

 

 

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