F(n) is non-prime for all p > 5. This is because F is the repeating portion of the decimal for 1/p.
For instance, F(7) = 142857
Necessarily F(7) * 7 = 999999.
Because 7 is not the only prime factor of 999999 (as 3 is also a factor), 142857 is not prime.
In general, for any prime > 3,
F(p)*p = 999..999 (with n nines).
p is not divisible by 3, so F(p) is necessarily divisible by 9.
q.e.d.
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by the way, why do 2, 3, and 5 not have this property?
F(3) = 3, which is prime, because 3 is divisible by 3.
F(2) arguably = 5 and F(5) arguably = 2. These do not follow the rule because 2 and 5 both divide 10, so F(2)*2 = F(5)*5 =10, not 9.
Edited on June 28, 2014, 9:35 pm
Solution (spoiler)
