A Pythagorean triple consists of three positive integers $a,b,c$ such that $$a^2+b^2=c^2.$$ There are infinitely many solutions. On the other hand, it is known that $$a^4+b^4=c^4$$ has no positive integer solutions.
However, I tried to modify the last equation and came up with the following question:
Are there positive integers $a,b,c,d$ such that they are all distinct and $$a^4+b^4=c^4+d^4?$$ If so, does it have infinitely many solutions? Does this equation even have a name? If so, a reference would be useful.