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A question about quadratic residues

  1. Mar 17, 2012 #1
    I need to prove that a be a odd integer that congruence X^2[itex]\equiv[/itex]a mod 2
    is always solvable with exactly one incongruent solution modulo 2.
    this question is linked with (b) let a be an odd integer. Prove that the congruence X^2[itex]\equiv[/itex]a mod 4 is solvable iff a[itex]\equiv[/itex]1 mod 4. in this case ,prove that X^2[itex]\equiv[/itex]a mod 4solutions has exactly two incongruent
    solutions modulo 4.

    these two seem to link with each other. And the proposition I learn is X^2[itex]\equiv[/itex]a mod p has either no solution or two solutions, but p there is an odd prime number. HOw to apply to the queations above?
  2. jcsd
  3. Mar 17, 2012 #2


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    Staff Emeritus
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    Gold Member

    You should be able to just check these by hand. For example does x^2=2 (mod 4) have any solutions? Just plug in 0,1,2,3 for x and see what you get
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