Timeline for Pythagorean triples and quadratic residues modulo primes

7 events

when toggle format	what		by	license	comment
May 17, 2023 at 16:14	history	edited	Martin Sleziak		edited tags
May 17, 2023 at 16:10	answer	added	Maciej Ulas		timeline score: 5
May 10, 2021 at 14:58	comment	added	Zhi-Wei Sun		I conjecture further that for any prime $p>828$ there is a positive integer $a\le\sqrt{p-2}$ such that $a^2-1,2a,a^2+1$ are all quadratic residues modulo $p$. Note that $(a^2-1)^2+(2a)^2=(a^2+1)^2$.
May 9, 2021 at 15:22	comment	added	Zhi-Wei Sun		By Fermat's last theorem, there are no positive squares $a,b,c$ satisfying $a^2+b^2=c^2$. This makes the question particularly interesting.
May 8, 2021 at 15:14	comment	added	Dr. Pi		The circle method does not work in three variables, at least when you have congruences. See the Crelle paper (new version of circle method) of Heath-Brown (comments after Theorem 8).
May 7, 2021 at 11:06	comment	added	George Shakan		For large primes you can use the circle method. See Corollary 4.15 in Tao and Vu's book on additive combinatorics for an easier variant, which may be modified.
May 6, 2021 at 22:02	history	asked	Zhi-Wei Sun	CC BY-SA 4.0