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Old 2007-10-22, 15:09   #1
Peter Hackman
 
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Oct 2007
linköping, sweden

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Default primtive roots mod p^2

p=40487 is a prime number, 5 is its least positive primitve root. It is not a primitive root modulo p^2 (the least primtive root mod p^2 is 10).

p is the smallest odd prime having this property. The next smallest example is q=6692367337, found by A Paszkiewicz, possibly others.

I know of no other example. Is anything known on this topic? I'm not even sure of its mathematical significance.
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Old 2007-10-22, 15:13   #2
R.D. Silverman
 
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Nov 2003

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Quote:
Originally Posted by Peter Hackman View Post
p=40487 is a prime number, 5 is its least positive primitve root. It is not a primitive root modulo p^2 (the least primtive root mod p^2 is 10).

p is the smallest odd prime having this property. The next smallest example is q=6692367337, found by A Paszkiewicz, possibly others.

I know of no other example. Is anything known on this topic? I'm not even sure of its mathematical significance.
On rough, probabilistic heuristics we should expect the number of
such primes less than N to be about loglog N. There should be infinitely many,
but they will be very very sparse.
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Old 2007-10-24, 06:41   #3
maxal
 
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Feb 2005

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http://www.research.att.com/~njas/sequences/A055578
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