Gauss’s Criterion

May 1, 2018

Yesterday was the 241st anniversary of the birth of Carl Gauss, who is perhaps the greatest mathematician who ever lived. In his honor, I picked a small task based on some mathematics that he discovered. Gauss’s Criterion states:

Let p be an odd prime and b a positive integer not divisible by p. Then for each positive integer 2 k − 1 < p, let rk be rk ≡ ( 2 k − 1) b (mod p) with 0 < rk < p, and let t be the number of even rk s. Then (b/p) = (-1)t, where (b/p) is the Legendre symbol.

Your task is to write a program to compute Gauss’s criterion, and confirm that it is the appropriate power of the Legendre symbol. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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One Response to “Gauss’s Criterion”

  1. Daniel said

    Here’s a solution in Python 2.

    def legendre(b, p):
      l = (b ** ((p - 1) / 2)) % p
      if l == p - 1:
        l = -1
      return l
    
    
    def gauss(b, p):
      r = ((x * b) % p for x in xrange(1, p, 2))
      t = sum(1 for x in r if (x % 2) == 0)
      return (-1) ** t
    
    
    ps = [3, 5, 7, 11, 13, 17, 19, 23, 29]
    for p in ps:
      for b in range(1, p):
        assert gauss(b, p) == legendre(b, p)
    

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