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.
* * *
