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I note that $$2(n^2+n+1)^2 -1= n^4+(n+1)^4.$$ This leads me to pose the following question.

Question 1. Are there infinitely many positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\mathbb N=\{0,1,\ldots\}$ with $|x-y|>1$?

I have found totally $53$ such positive integers $m$ below $10^8$, the list of them is available from http://oeis.org/A343913 . Here I give a new larger example: $$2\times188717259^2-1=15335^4+11234^4. $$ I guess that there are infinitely many such positive integers $m$.

I also observe that $$2(n^2+3)^2-2^4= (n+1)^4+(n-1)^4\ \ \mbox{with}\ n+1-(n-1)=2.$$ I have the following question analogous to Question 1.

Question 2. Are there infinitely many positive integers $m$ such that $2m^2-2^4 =x^4+y^4$ for some $x,y\in\mathbb N$ with $|x-y|>2$?

I have found totally $62$ such positive integers $m$ not exceeding $10^8$, the list of them is available from http://oeis.org/A343917 . For example, $$2\times97077407^2-2^4=11563^4+5583^4.$$ I also guess that Question 2 has a positve answer.

Your comments are welcome!

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  • $\begingroup$ You can conjecture for $2m^2-a^4=x^4+y^4$ using a famous identity $a^4+b^4+(a+b)^4=2(a^2+ab+b^2)^2$ $\endgroup$
    – Tomita
    Commented May 4, 2021 at 23:17

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