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Old 2017-04-20, 15:15   #50
Dr Sardonicus
 
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Feb 2017
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I have no idea of any significance to numbers of the form 10^n + 7.

The polynomial x^n + 7 is irreducible in Q[x] for every positive integer n, by Eisenstein's criterion with p = 7.

FWIW, I checked 10^n + 7 up to the limit n = 2000, and found pseudoprimes for the exponents

n = 1, 2, 4, 8, 9, 24, 60, 110, 134, 222, 412, 700, 999, and 1383.

I don't see any obvious pattern. Most (but not all) of these n are even. Six of them are divisible by 3.

If n is odd, then 10*(10^n + 7) = (10^(n+1)/2)^2 + 70, so -70 is a quadratic residue of every prime factor of 10^n + 7. If one is testing small primes as divisors for the given odd value of n, this would eliminate about half the candidates.
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