p^6 - 7p^2 + 6 is a multiple of 3. p^2 = 1 mod 3 for all primes > 3, and so therefore is p^6 = (p^2)^3. So the first term is 1 mod 3, the second term is -1 mod 3 (since 7 == 1 mod p), and the third term is zero mod 3
p^6 = 1 mod 7 for all primes > 7 (i.e. beginning with 11) since phi(7) = 6. Then the first term is 1 mod 7 the second term is 0 mod 7 and the third is -1 mod 7 so the expression is a multiple of 7.
If we write p = 4k +- 1 and expand (4k+-1)^6 -7(4k+-1)^2 + 6, we get an ugly polynomial but all of the coefficients are multiples of 32 and not all of them are multiples o 64. So the expression is a multiple of 32
The expression, then, is a multiple of 32*3*7 = 672 for all p > 7 (or >= 11)
Can there be other factors? No.
When p = 11, the expression is 1770720 which is 672 * (5*17*31)
while when p = 13, the expression is 4825632, which is 672*( 43*167). The GCD of these two number is obviously 672, so the GCD of any set which includes both cannot exceed 672. Since the expression is a multiple of 672 for all p >= 11 and since the first two elements have no other common factors, the GCD must be 672.
Analytic solution
