*** See Edit section below ***
I saw quite a few questions with this phrase even in the title, but they are all about squares if I'm correct. Of course I know this isn't really the Pythagorean theorem, just resembles it. But then what is this? I don't even know the correct tags I should use.
In $3$ dimensions and in a $2000 \times 2000 \times 2000$ cube I found $7448$ number quadruplets for which this is true:
$$a^3+ b^3 + c^3 = d^3$$
For example $607$, $1185$, $1284$, and $1588$.
Similarly, in $4$ dimensions and in a $1500 \times 1500 \times 1500 \times 1500$ space I found only $6$ quintuplets that satisfy this:
$$a^4 + b^4 + c^4 + d^4 = e^4$$
They are:
30,120,272,315,353
60,240,544,630,706
90,360,816,945,1059
120,480,1088,1260,1412
240,340,430,599,651
480,680,860,1198,1302
I'm running this for $5$ dimensions but I'm expecting this to take a few months. One week in and still below $5$% of the work. So far I've got this:
2,298,351,474,500,575
4,26,139,296,412,427
4,596,702,948,1000,1150
7,43,57,80,100,107
8,52,278,592,824,854
8,170,367,689,811,875
14,86,114,160,200,214
14,95,545,586,644,744
15,32,375,789,933,1004
15,260,464,786,890,975
19,43,46,47,67,72
19,201,347,388,448,503
21,23,37,79,84,94
21,129,171,240,300,321
27,106,388,485,604,650
28,172,228,320,400,428
31,105,139,314,416,435
35,215,285,400,500,535
38,86,92,94,134,144
Indeed, $35^5 + 215^5 + 285^5 + 400^5 + 500^5 = 535^5$
Any help would be greatly appreciated. Does this have a name? Are there people studying this?
I started a repository with my findings and what my stupid brain thinks about them here. Full list of solutions in 3 dimensions is this file.
Edit:
This is closed. My first observation may stand, but the second is not true.
As Robert Lee in a comment here and some Reddit users over there pointed out, my second observation is basically Euler's sum of powers conjecture. And It's been disproved already in 1966 if I'm correct. For example:
$$95800^4 + 217519^4 + 414560^4 = 422481^4$$
As for the first observation, here was a comment (later deleted) that said this isn't related to the Pythagorean theorem at all. This is possible, I'm no mathematician. But how about this interpretation:
Dropping the triangles, it says that you can find two numbers, raise them to the power of two. Then find a 3rd one, and the sum of two squares will be the square of the third's square. Doesn't work for any number but you can find infinitely many combinations. I'm saying you can extend this, and you can find three numbers, raise them to the power of three, then find a fourth one ... etc.
(Deleted the links)