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Number Theory: Pigeonhole Principle

  1. Jun 4, 2009 #1
    2exmt6c.jpg

    What have I done wrong here?

    Define f: A -> B

    (a,b) -> f(a,b) ≡ a + bc mod p

    Let A = {(a,b): a,b integers, 0≤a,b≤√p}

    B = {0,1,2,..,p-1}

    By pigeonhole principle there are distinct (a1,b1), (a2,b2) in A with f(a1,b1) = f(a2,b2).


    => a1+b1c ≡ a2+b2c mod p

    => (a1-a2) ≡ (b2-b1)c mod p

    => (a1-a2)2 ≡ (b2-b1)2c2 ≡ (b2-b1)2(-2) mod p

    Let a = a1 - a2 and b = b1-b2

    => a2 ≡ -2b2 mod p

    => a2+2b2 is a multiple of p

    As (a1,b1) ≠ (a2,b2)

    (a,b) ≠ (0,0) so a2+b2 > 0

    As 0≤a1≤√p and 0≤a2≤√p

    then

    -√p < a1 - a2 < √p

    i.e. -√p < a < √p => a2 < p

    Similarly b2 < p => 2b2 < 2p

    => a2+2b2 < p + 2p = 3p

    So a2+2b2 is a multiple of p strictly between 0 and 3p

    So a2+2b2 = p

    OR a2+2b2 = 2p surely?
     
  2. jcsd
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