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%I #4 Jun 29 2008 03:00:00
%S 3,11,3511,6692367337
%N Primes p for which the least-magnitude negative primitive root is not a primitive root of p^2. Like A055578, but for negative rather than positive primitive roots.
%C There is a rough heuristic suggesting that a prime p will occur in this list with probability 1/p; the actual density seen here tails off faster than that. No other primes with this property exist up to 2^36. Used for testing a multiprecision division algorithm.
%C The sequence giving the least-magnitude primitive roots r of primes p for which r is not a primitive root of p^2 begins -1,-3,-2,-5,..., with no other cases known up to 2^36.
%e -3 is a primitive root of 11. That is, the successive powers of -3 work through all the nonzero residues modulo 11 before coming round through 1 to -3 again: -3, -2, -5, 4, -1, 3, 2, 5, -4, 1, -3, ...
%e -3 also happens to be the negative number of least magnitude with this property (-1 obviously fails, -2 yields -2, 4, 3, 5, 1, -2 ...) Modulo 11^2 = 121, however, successive powers of -3 do not yield all the corresponding residues (that is, all the ones which aren't multiples of 11): we only get -3, 9, -27, 81, -1, 3, -9, 27, -81, 1, -3, ...
%Y Cf. A055578, A060503. A060504.
%K hard,nonn
%O 1,1
%A Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Dec 13 2004
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