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Distribution of square residuals mod prime

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  • Kermit Rose
    If p is a positive prime equal to 1 mod 4, then each square residual mod p is the difference of non-zero squares mod p in (p-5)/4 ways, and each non-square
    Message 1 of 1 , Jul 17 10:38 AM
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      If p is a positive prime equal to 1 mod 4,

      then each square residual mod p
      is the difference of non-zero squares mod p
      in (p-5)/4 ways,

      and

      each non-square residual mod p,
      is the difference of non-zero squares mod p
      in (p-1)/4 ways.

      This is consistent with
      Number of nonzero square residual pairs with non square residual difference
      + number of nonzero square residual pairs with square residual difference
      + number of nonzero square residual pairs with zero difference
      = ((p-1)/2) **2


      If p = 3 mod 4,
      then
      each nonzero element mod p
      is the difference of non-zero squares mod p
      in (p-3)/4 ways.

      This is consistent with


      Number of nonzero square residual pairs with non square residual difference
      + number of nonzero square residual pairs with square residual difference
      + number of nonzero square residual pairs with zero difference
      = ((p-1)/2) **2


      Kermit Rose
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