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

In summary, to show that the polynomial g(x)=(x^2 -5) (x^2-41)(x^2-205) has a solution modulo any integer n∈ℕ, we need to consider quadratic residues and determine if any of the numbers 5, 41, or 41*205 are quadratic residues mod n. This can be done using Legendre's Symbol and other resources such as Wolfram. Additionally, it is important to note that at least one of these numbers must be a quadratic residue mod n for each n. Starting with small cases and considering what must be true for a number to be a root of g, mod n can also be helpful.
  • #1
qaz
7
0
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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  • #2
you mean root; if there is a solution it should come from considering quadratic residues, i imagine.
 
  • #3
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.
 
  • #4
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?
 
  • #5
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.
 
  • #6
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.
 
  • #7
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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