Can the Polynomial g(x) Have a Solution Modulo Any Integer n?

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Discussion Overview

The discussion centers around the polynomial g(x) = (x^2 - 5)(x^2 - 41)(x^2 - 205) and whether it has a solution modulo any integer n in the natural numbers. Participants explore concepts related to quadratic residues and the implications of specific integer values.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant requests to show that the polynomial g(x) has a solution modulo any integer n.
  • Another participant suggests that if there is a solution, it should be considered in the context of quadratic residues.
  • A participant expresses difficulty in progressing with the problem due to inadequate resources and explanations in their class.
  • Another participant encourages starting with the conditions necessary for a number m to be a root of g modulo n, and what must hold if no such m exists.
  • One participant points out the significance of the number 205 and introduces the concept of Legendre's Symbol as potentially useful.
  • A participant notes that 205 can be expressed in terms of quadratic residues, leading to two cases for the problem based on the values of 5 and 41.
  • Another participant acknowledges the previous point but emphasizes the need for a clearer explanation, stating that at least one of the numbers 5, 41, or 41*205 must be a quadratic residue mod n for each n.

Areas of Agreement / Disagreement

The discussion contains multiple competing views and remains unresolved, with participants exploring different aspects of the problem without reaching a consensus.

Contextual Notes

Participants have not fully resolved the implications of quadratic residues or the conditions under which the polynomial has solutions, and there are missing assumptions regarding the nature of n.

qaz
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show that the polynomial [itex]g(x)=(x^2 -5) (x^2-41)(x^2-205)[/itex] has a solution modulo any integer [itex]n[/itex]∈ℕ.
 
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you mean root; if there is a solution it should come from considering quadratic residues, i imagine.
 
ok, but i am stil stuck i don't know where to go from here...we don't have a good book for this class and it wasnt explained well at all.
 
Have you done anything? (whether successful or not)

Sometimes starting small helps. What has to be true for some number m to be a root of g, mod n? What has to be true if no such m exists?
 
Have you also noticed something important about 205?

What do you konw about Legendre's Symbol? I'm sure there are lots of useful resources out there, try Wolfram.
 
ok, i know that 205 can be written as the following: (205/n)=(5/n)(41/n), which reduces to =(-1)(-1)=1. so there are either 2 cases for this problem, (5/n)=1 or (41/n)=1.
 
that is possibly correct in spirit, but needs better explanation: at least one of 5,41 or 41*205 must be a quadratic residue mod n for each n because...
 

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