Prove that if a is a quadratic residue mod an odd prime p th

Prove that if a is a quadratic residue mod an odd prime p, then a is not a primitive root mod p.

Solution

If a is a quadratic residue modulo the odd prime p:

By Euler’s criterion it means that a^{( p 1) / 2} 1 (mod p )

So, from this it leads to that the order of a modulo p is less that p 1 = ( p ),

Hence \'a\' cannot be a primitive root modulo p.

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