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.
