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COMMENT
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It is conjectured that the sequence of Fermat primes (A019434) is complete; if so then this sequence is complete:
Suppose that x is a positive integer for which x^x+1 is prime. If x has an odd prime factor p, then x^x + 1 = (x^(x/p))^p + 1 is divisible by x^(x/p) + 1, so it is not prime. So x must be a power of 2. Hence x^x is also a power of 2, so x^x+1 is a Fermat prime.
If there are no Fermat primes beyond the known ones (as is widely believed), then x must be 1, 2, or 4. Letting x=phi(n), it is easy to see that n must be one of the numbers listed. - Dean Hickerson dean(AT)math.ucdavis.edu, Feb 11, 2002
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