Fermat prime greater than three implies three is primitive root

From Number
Revision as of 19:16, 3 January 2012 by Vipul (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search


Suppose is a positive integer (in particular, it is not zero) such that the Fermat number is a prime number (and hence a Fermat prime). Then, the number 3 is a quadratic nonresidue modulo , and hence a primitive root modulo .

Related facts

Facts used

  1. Quadratic reciprocity
  2. Quadratic nonresidue equals primitive root for Fermat prime


We use Fact (1), which says that, in terms of Legendre symbols:

Note that:

  • Since , , so the exponent on the right side is even, so the right side is 1.
  • We know that , so , so . Thus, which is -1.

Plugging these in, we get:

Thus, 3 is a quadratic nonresidue mod . Combine with Fact (2) to obtain that it is a primitive root.