Timeline for The modular arithmetic contradiction trick for Diophantine equations
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2023 at 7:10 | answer | added | user178594 | timeline score: 5 | |
| Jun 21, 2013 at 16:41 | comment | added | Daniel Loughran | I forgot to add that with regards to the Hasse principle, one also asks for solubility at the "infinite prime", which corresponds to asking for solutions in $\mathbb{R}$. | |
| Jun 21, 2013 at 11:58 | comment | added | Daniel Loughran | This is a lot more than a ``slick ad-hoc trick". The idea that the existence of an integer solution implies the existence of solutions modulo every $n$ is really at the cornerstone of the modern study of diophantine equations. Equations for which the converse also holds are said to satisfy the Hasse principle; as already noted this principle does not hold in general. In practice one often chooses an $n$ such that the equation has singular reduction mod $n$ (e.g. $n$ divides an exponent or a coefficient), since often one gets solutions for free for $n$ with non-singular reduction. | |
| Jun 21, 2013 at 10:51 | answer | added | KConrad | timeline score: 17 | |
| Jun 21, 2013 at 8:55 | comment | added | zeb | For Diophantine equations coming from curves, the Hasse-Weil bound shows that you can solve your Diophantine equation mod p for any sufficiently large prime p, and for higher dimensional varieties I suspect one can use the Weil conjectures to get a similar result. Furthermore, if the equation is solvable mod p, then usually Hensel's lemma allows you to lift the solution to a solution mod any power of p. So really there are only finitely many congruences one needs to check to see if a Diophantine equation can be ruled out by congruences. | |
| Jun 21, 2013 at 8:33 | answer | added | Alex B. | timeline score: 17 | |
| Jun 21, 2013 at 6:10 | answer | added | Gerry Myerson | timeline score: 25 | |
| Jun 21, 2013 at 6:01 | comment | added | Favst | I don't quite know why this has been down-voted. I asked a question that has bothered me since my old olympiad days, looking for a serious answer. And potentially interesting answers exist. | |
| Jun 21, 2013 at 5:50 | history | edited | Favst | CC BY-SA 3.0 |
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| Jun 21, 2013 at 5:28 | history | asked | Favst | CC BY-SA 3.0 |