Timeline for $x^4+y^4$ powerful for relatively prime $x,y$

6 events

when toggle format	what		by	license	comment
Jan 11, 2015 at 13:09	history	edited	John R Ramsden	CC BY-SA 3.0	minor tweaks
Jan 10, 2015 at 18:08	comment	added	John R Ramsden		I beefed up my suggested method with a few (obvious) extra checks, and this involves calculations of only GCDs, quotients, and square roots (not prime factorisations for example). But, as you say, the search space is very large. So probably a more sophisticated approach is required.
S Jan 10, 2015 at 18:01	history	suggested	CommunityBot	CC BY-SA 3.0	Removed unsatisfactory final comment, and beefed up "method 2"
Jan 10, 2015 at 17:41	review	Suggested edits
S Jan 10, 2015 at 18:01
Jan 10, 2015 at 3:53	comment	added	Noam D. Elkies		In fact if $\gcd(x,y) = 1$ then every odd prime factor of $x^4 + y^4$ is congruent to $1 \bmod 8$, not just $1 \bmod 4$ (and if $x^4+y^4$ is powerful then it must be odd). That does cut down the search space a bit $-$ a number of bits, even $-$ but still not nearly enough to make it feasible to provably find all examples with $x^4 + y^4 < 3 \cdot 10^{36}$. It is, however, one ingredient of the closest thing I have to a feasible strategy, which I may post as a partial answer to my own question if nothing better appears here soon.
Jan 9, 2015 at 15:34	history	answered	John R Ramsden	CC BY-SA 3.0