I used a GCD algorithm on n^5 + 5 and (n+1)^5+5 = n^5 + 5n^4 + 10n^3 + 10n^2 + 5n + 6.
P1 = n^5 + 5n^4 + 10n^3 + 10n^2 + 5n + 6
P2 = n^5 + 5
P3 = P1 - P2 = 5n^4 + 10n^3 + 10n^2 + 5n + 1
P4 = 5*P2 - n*P3 = -10n^4 - 10n^3 - 5n^2 - n + 25
P5 = 2*P3 + P4 = 10n^3 + 15n^2 + 9n + 27
P6 = P4 + n*P5 = 5n^3 + 4n^2 + 26n + 25
P7 = P5 - 2*P6 = 7n^2 - 43n - 23
P8 = 7*P6 - 5n*P5 = 243n^2 + 297n + 175
P9 = 243*P7 - 7*P8 = -12528n - 6814
P10 = 464*P8 + 9n*P9 = 76482n + 81200
P11 = 4249*P9 + 696*P10 = 27562514
Any possible common factor m must be a factor of 27562514 = 2 * 7 * 1968751. Checking powers of 5 mod 2 and mod 7 finds those mods don't work so any possible nontrivial common factor of m must be 1968751. Charlie's computer search shows that 1968751 is a valid value for m.
Potential solution
