5 ms·

> “If there are no solutions modulo a prime, then you know there are no solutions,” Can anyone point me towards a proof of this?

• mkbosmans 5y ago
  Assume there is a solution to the equation. Then if you take the mod n of both sides the new equality will also hold, for any n. Therefore, if you can prove there are no solution for the mod n equation, then the original equation is also not solvable. This of course also hold when n is prime.
  • deleted 5y ago
    [deleted]
• SamReidHughes 5y ago
  If there are solutions, they work modulo any prime.
• deleted 5y ago
  [deleted]
• qsort 5y ago
  The wikipedia page is rather comprehensive: https://en.wikipedia.org/wiki/Chinese_remainder_theorem https://en.wikipedia.org/wiki/Chinese_remainder_theorem It also covers the straightforward generalization of the theorem to principal ideal domains (e.g. polynomials over a field)
• woko 5y ago
  This quote is specifically about Diophantine equations. You can find a similar statement as "Proposition 3.2.2" in Chapter 3 "Modular Arithmetic" [1] of this introductory course [2] to Number Theory. Here is a picture of the statement [3]. Basically, if there exists a number n such that there is no solution modulo n, then you know there are no solutions. Here is a simple application [4] of the proposition with n = 2 and 3. [1] <http://www2.math.ou.edu/~kmartin/intro-nt/ch3.pdf http://www2.math.ou.edu/~kmartin/intro-nt/ch3.pdf> [2] <http://www2.math.ou.edu/~kmartin/intro-nt/ http://www2.math.ou.edu/~kmartin/intro-nt/> [3] <https://i.ibb.co/8mNHW0Q/2021-09-20-10-34-25.png https://i.ibb.co/8mNHW0Q/2021-09-20-10-34-25.png> [4] <https://i.ibb.co/Qffr9k2/2021-09-20-10-56-29.png https://i.ibb.co/Qffr9k2/2021-09-20-10-56-29.png>