<s>[INST] Small powers of small numbers: Is there really a pattern or am I just used to looking for them? This probably is just a coincidence, but I've always found it interesting and I wanted to put some feelers out there to see if maybe there really is something to it after all.
There are these collections of powers, and in one case a sum of powers in two different ways, that have similar digits to each other. 
$$3^7 = 2187$$
$$6^4 = 1296$$
$$12^3 = 1728$$
$$13^3 = 2197$$
$$1^3 + 12^3 = 9^3 + 10^3 = 1729$$
$$3^6 = 729$$
(I suppose the last two are inevitable in any radix.)
We also have
$$2^{10} = 1024$$
$$7^4 = 2401$$
and
$$2^8 = 256$$
$$5^4 = 625$$
$$24^2 = 576$$
I can see the following class is inevitable in any radix > 4.
$$12^2 = 144$$
$$21^2 = 441$$
$$13^2 = 169$$
$$31^2 = 961$$
Is there an overarching mechanism at play here, or am I just finding statistically insignificant patterns?
 [/INST] This is really only a tiny piece of an answer, but...
This is a very interesting question. My first reaction is to think that it can't be a coincidence, but I'm not certain.
Now, a statistical explanation seems more or less plausible. You're mostly looking at numbers up to $13$ raised to numbers up to $10$, so there are about $12 \cdot 9 = 108$ "interesting" powers. Meanwhile, there are $286$ ways of choosing $4$ digits, with duplicates allowed but disregarding order. So it shouldn't be too uncommon for powers to be anagrams.
Still, this "family" of prime powers seems pretty compelling:
$5^2 = 25 = 24 + 1$
$7^4 = 2401 = 24 \cdot 10^2 + 1$
$2^{10} = 1024 = 24 + 10^3$
We can examine the relationships between these equations a little more clearly by pairing them up, multiplying by $10^2$ as needed, and subtracting them to cancel the $24$s:
$2^2 \cdot 5^4 - 7^4 = 10^2 - 1$
$2^{10} - 5^2 = 10^3 - 1$
$2^{12} \cdot 5^2 - 7^4 = 10^5 - 1$
By coincidence or otherwise, all of the powers on the left-hand side are even, so we can apply the difference-of-squares formula. We can also factor the right-hand sides, of course. These equations factor as:
$(2 \cdot 5^2 - 7^2) (2 \cdot 5^2 + 7^2) = (10 - 1) (10 + 1)$
$(2^5 - 5) (2^5 + 5) = (10 - 1) (10^2 + 10 + 1)$
$(2^6 \cdot 5 - 7^2) (2^6 \cdot 5 + 7^2) = (10 - 1) (10^4 + 10^3 + 10^2 + 10 + 1)$
Then, if we replace each factor with its own prime factorization, we end up with:
$(1) (3^2 \cdot 11) = (3^2) (11)$
$(3^3) (37) = (3^2) (3 \cdot 37)$
$(271) (3^2 \cdot 41) = (3^2) (41 \cdot 271)$
But this definitely doesn't seem like a compelling explanation of anything. We started with some pretty interesting equations, like $2^{10} - 5^2 = 10^3 - 1$, and "explained" them by progressively factoring them, but there's no explanation of why the prime factors on the left ended up matching the prime factors on the right.
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