MathOverflow

How can I find the general solution of $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)?And how did Euler find the solution $158^4+59^4=133^4+134^4$?

Let $E \subset \mathbb R^n$ have nonzero finite $s$-dimensional Hausdorff measure. Consider the natural uniform probability measure on $E$, which is just normalised $s$-Hausdorff measure.
Question: Let $X, Y$ be drawn uniformly and independently from $E$. Is it true there exists a constant $\delta(s, n) > 0$ independent of $E$ such that...

Suppose I have two strings $s_1$ and $s_2$ of equal length $L$ with an alphabet size of $k \geq 2$. Suppose further that these two strings initially have a Hamming distance equal to $d_0 = H(s_1,s_2)$.
I then proceed through each character in one of the strings (say $s_2$), changing with probability $\mu$ the character to another distinct value in the alphabet, uniformly from the $k-1$ choices remaining. The number of edits is therefore distributed...

I am trying to understand whether there is any existing machinery that could plausibly give cancellation in the following sparse divisor/Kloosterman-type regime.
Let
$$R(h)=h^6+27k^2,$$