$3^2 + 4^2 = 5^2$ and $3^3 + 4^3 + 5^3 = 6^3$: Is this just a coincidence?

Question

I noticed a curious pattern involving the numbers 3, 4, 5, and 6:

$3^2+4^2=5^2$

$3^3+4^3+5^3=6^3$

I don't think I have to explain the pattern here. But there is a high chance for this to be just a coincidence because these are the only instances I've found where this pattern occurs.

Is there any deeper mathematical significance behind this pattern, or is it just a coincidence? I understand that coincidences can happen in mathematics, but I wonder if there is a reason behind why this occurs here. Is there a more general principle or theory that can explain why these specific numbers satisfy these equations?

I’m curious if there's something interesting going on or if my question doesn't have much significance. I appreciate any insights into why this pattern appears or if it connects to other mathematical concepts. I know the brilliant minds on Math Exchange always have beautiful answers for every kind of question, even if the question seems a bit silly. Please help me with my curiosity.

I think this is just a neat number property, unless a deeper pattern is found. For example, $3^4 + 4^4 + 5^4 + 6^4 ≠ 7^4$. — keixx, Oct 20 '24 at 03:02
@keixx I think I have to agree with you. Though it is a little bit disappointing to believe that there is no pattern. Thank you for your idea — Ruchin Himasha, Oct 20 '24 at 03:06
Define coincidence :-) Those are coincidencies unless/until a pattern is found. Like $0+3+5+6=1+2+4+7$ as well as $0^2+3^2+5^2+6^2=1^2+2^2+4^2+7^2$. Which can be extended to two complementary sets such that their sums of 1st, 2nd as well as 3rd powers agree. — Jyrki Lahtonen, Oct 20 '24 at 03:09
The strong law of small numbers: there aren't enough small numbers to avoid coincidences. — Arturo Magidin, Oct 20 '24 at 03:30
This question is similar to: Why $3^{3}+4^{3}+5^{3}$ is equal $6^{3}$ and with different numbers length too? — Martin Brandenburg, Oct 20 '24 at 03:55
I think the pattern is a coincidnce. A quick search shows trivial solutions like $\quad(-1)^{3}+0^{3}+1^{3}=0^{3}\quad$ and a non-trivial$\quad1^{3}+6^{3}+8^{3}=9^{3}.\quad$
I once wrote a program that revealed more but, other than $,{3,4,5,6},,$ I do not know of any other consecutive number solutions to $\quad x^y+(x+1)^y+\cdots+(x+n-1)^y=(x+n)^y$. — poetasis, Oct 20 '24 at 13:16

$3^2 + 4^2 = 5^2$ and $3^3 + 4^3 + 5^3 = 6^3$: Is this just a coincidence?

0 Answers0