Great common divisor with powers of linear polynomials

# The following code is written in the language GAP: https://www.gap-system.org/
# See https://mathoverflow.net/questions/352962 for more context.


# computes the list of primes p such that p - 1 divides n - 1
# equivalently, the prime field F_p is an n-ring.

myPrimes := function(n)
	local L,d;
	L := [];
	for d in DivisorsInt(n-1) do
		if IsPrimeInt(d+1) then
			Add(L,d+1);
		fi;
	od;
	return L;
end;

# checks if n - 1 is not divisible by any p^k - 1 with k > 1;
# we then say that n is special at p.
# this is equivalent to saying that F_p is the only n-field of char. p.

isSpecialAt := function(n,p)
	local k;
	for k in [2..n] do
		if (n-1) mod (p^k-1) = 0 then
			return false;
		fi;
	od;
	return true;
end;

# checks for all s <= n <= t and all odd primes p for which n is special
# if the gcd of all (x+i)^n - (x+i) for i=0,..,p-1 is x^p - x.
# also prints out the first i when the gcd has been found (often it's 1).

testMe := function(s,t)
	local i,x,n,p,g,found;
	for n in [s..t] do
		for p in myPrimes(n) do
			if isSpecialAt(n,p) and p > 2 then
				Print([n,p]);
				x := Indeterminate(GF(p),"x");
				g := x^n - x;
				found := false;
				for i in [1..p-1] do
					g := Gcd(g,(x+i)^n - (x+i));
					if (g = x^p - x) then
						found := true;
						Print(" ok at ",i,"\n");
						break;
					fi;
				od;
				if (found = false) then
					Print(" not ok!\n");
				fi;
			fi;
		od;
	od;
end;


# Here is the result of the test function:

gap> testMe(2,200);
[ 3, 3 ] ok at 1
[ 5, 3 ] ok at 1
[ 5, 5 ] ok at 1
[ 7, 3 ] ok at 2
[ 7, 7 ] ok at 1
[ 9, 5 ] ok at 1
[ 11, 3 ] ok at 1
[ 11, 11 ] ok at 1
[ 13, 3 ] ok at 2
[ 13, 5 ] ok at 2
[ 13, 7 ] ok at 1
[ 13, 13 ] ok at 1
[ 15, 3 ] ok at 1
[ 17, 5 ] ok at 1
[ 17, 17 ] ok at 1
[ 19, 3 ] ok at 2
[ 19, 7 ] ok at 1
[ 19, 19 ] ok at 1
[ 21, 3 ] ok at 1
[ 21, 5 ] ok at 4
[ 21, 11 ] ok at 1
[ 23, 3 ] ok at 1
[ 23, 23 ] ok at 1
[ 25, 7 ] ok at 3
[ 25, 13 ] ok at 1
[ 29, 3 ] ok at 1
[ 29, 5 ] ok at 1
[ 29, 29 ] ok at 1
[ 31, 3 ] ok at 2
[ 31, 7 ] ok at 1
[ 31, 11 ] ok at 2
[ 31, 31 ] ok at 1
[ 33, 5 ] ok at 1
[ 33, 17 ] ok at 1
[ 35, 3 ] ok at 1
[ 37, 3 ] ok at 2
[ 37, 5 ] ok at 2
[ 37, 7 ] ok at 1
[ 37, 13 ] ok at 1
[ 37, 19 ] ok at 1
[ 37, 37 ] ok at 1
[ 39, 3 ] ok at 1
[ 41, 5 ] ok at 4
[ 41, 11 ] ok at 3
[ 41, 41 ] ok at 1
[ 43, 3 ] ok at 2
[ 43, 7 ] ok at 6
[ 43, 43 ] ok at 1
[ 45, 3 ] ok at 1
[ 45, 5 ] ok at 1
[ 45, 23 ] ok at 1
[ 47, 3 ] ok at 1
[ 47, 47 ] ok at 1
[ 49, 13 ] ok at 1
[ 49, 17 ] ok at 2
[ 51, 3 ] ok at 1
[ 51, 11 ] ok at 1
[ 53, 5 ] ok at 1
[ 53, 53 ] ok at 1
[ 55, 3 ] ok at 2
[ 55, 7 ] ok at 1
[ 55, 19 ] ok at 1
[ 57, 5 ] ok at 1
[ 57, 29 ] ok at 1
[ 59, 3 ] ok at 1
[ 59, 59 ] ok at 1
[ 61, 3 ] ok at 2
[ 61, 5 ] ok at 4
[ 61, 7 ] ok at 1
[ 61, 11 ] ok at 4
[ 61, 13 ] ok at 1
[ 61, 31 ] ok at 1
[ 61, 61 ] ok at 1
[ 63, 3 ] ok at 1
[ 65, 5 ] ok at 1
[ 65, 17 ] ok at 1
[ 67, 3 ] ok at 2
[ 67, 7 ] ok at 1
[ 67, 23 ] ok at 2
[ 67, 67 ] ok at 1
[ 69, 3 ] ok at 1
[ 69, 5 ] ok at 1
[ 71, 3 ] ok at 1
[ 71, 11 ] ok at 1
[ 71, 71 ] ok at 1
[ 73, 7 ] ok at 3
[ 73, 13 ] ok at 1
[ 73, 19 ] ok at 3
[ 73, 37 ] ok at 1
[ 73, 73 ] ok at 1
[ 75, 3 ] ok at 1
[ 77, 3 ] ok at 1
[ 77, 5 ] ok at 1
[ 79, 7 ] ok at 1
[ 79, 79 ] ok at 1
[ 81, 5 ] ok at 4
[ 81, 11 ] ok at 3
[ 81, 17 ] ok at 1
[ 81, 41 ] ok at 1
[ 83, 3 ] ok at 1
[ 83, 83 ] ok at 1
[ 85, 3 ] ok at 2
[ 85, 5 ] ok at 2
[ 85, 7 ] ok at 6
[ 85, 13 ] ok at 4
[ 85, 29 ] ok at 2
[ 85, 43 ] ok at 1
[ 87, 3 ] ok at 1
[ 89, 5 ] ok at 1
[ 89, 23 ] ok at 3
[ 89, 89 ] ok at 1
[ 91, 3 ] ok at 2
[ 91, 7 ] ok at 1
[ 91, 11 ] ok at 2
[ 91, 19 ] ok at 2
[ 91, 31 ] ok at 1
[ 93, 3 ] ok at 1
[ 93, 5 ] ok at 1
[ 93, 47 ] ok at 1
[ 95, 3 ] ok at 1
[ 97, 13 ] ok at 1
[ 97, 17 ] ok at 4
[ 97, 97 ] ok at 1
[ 99, 3 ] ok at 1
[ 101, 3 ] ok at 1
[ 101, 5 ] ok at 4
[ 101, 11 ] ok at 1
[ 101, 101 ] ok at 1
[ 103, 3 ] ok at 2
[ 103, 7 ] ok at 1
[ 103, 103 ] ok at 1
[ 105, 5 ] ok at 1
[ 105, 53 ] ok at 1
[ 107, 3 ] ok at 1
[ 107, 107 ] ok at 1
[ 109, 3 ] ok at 2
[ 109, 5 ] ok at 2
[ 109, 7 ] ok at 1
[ 109, 13 ] ok at 1
[ 109, 19 ] ok at 1
[ 109, 37 ] ok at 1
[ 109, 109 ] ok at 1
[ 111, 3 ] ok at 1
[ 111, 11 ] ok at 10
[ 111, 23 ] ok at 1
[ 113, 5 ] ok at 1
[ 113, 17 ] ok at 1
[ 113, 29 ] ok at 1
[ 113, 113 ] ok at 1
[ 115, 3 ] ok at 2
[ 115, 7 ] ok at 2
[ 117, 3 ] ok at 1
[ 117, 5 ] ok at 1
[ 117, 59 ] ok at 1
[ 119, 3 ] ok at 1
[ 121, 7 ] ok at 3
[ 121, 13 ] ok at 1
[ 121, 31 ] ok at 3
[ 121, 41 ] ok at 2
[ 121, 61 ] ok at 1
[ 123, 3 ] ok at 1
[ 125, 3 ] ok at 1
[ 127, 3 ] ok at 2
[ 127, 7 ] ok at 6
[ 127, 19 ] ok at 1
[ 127, 43 ] ok at 1
[ 127, 127 ] ok at 1
[ 129, 5 ] ok at 1
[ 129, 17 ] ok at 1
[ 131, 11 ] ok at 1
[ 131, 131 ] ok at 1
[ 133, 3 ] ok at 2
[ 133, 5 ] ok at 2
[ 133, 7 ] ok at 1
[ 133, 13 ] ok at 1
[ 133, 23 ] ok at 4
[ 133, 67 ] ok at 1
[ 135, 3 ] ok at 1
[ 137, 5 ] ok at 1
[ 137, 137 ] ok at 1
[ 139, 3 ] ok at 2
[ 139, 7 ] ok at 1
[ 139, 47 ] ok at 2
[ 139, 139 ] ok at 1
[ 141, 3 ] ok at 1
[ 141, 5 ] ok at 4
[ 141, 11 ] ok at 1
[ 141, 29 ] ok at 2
[ 141, 71 ] ok at 1
[ 143, 3 ] ok at 1
[ 145, 13 ] ok at 1
[ 145, 17 ] ok at 6
[ 145, 19 ] ok at 3
[ 145, 37 ] ok at 1
[ 145, 73 ] ok at 1
[ 147, 3 ] ok at 1
[ 149, 3 ] ok at 1
[ 149, 5 ] ok at 1
[ 149, 149 ] ok at 1
[ 151, 3 ] ok at 2
[ 151, 7 ] ok at 1
[ 151, 11 ] ok at 2
[ 151, 31 ] ok at 1
[ 151, 151 ] ok at 1
[ 153, 5 ] ok at 1
[ 155, 3 ] ok at 1
[ 155, 23 ] ok at 1
[ 157, 5 ] ok at 3
[ 157, 7 ] ok at 1
[ 157, 13 ] ok at 12
[ 157, 53 ] ok at 2
[ 157, 79 ] ok at 1
[ 157, 157 ] ok at 1
[ 159, 3 ] ok at 1
[ 161, 5 ] ok at 4
[ 161, 11 ] ok at 3
[ 161, 17 ] ok at 1
[ 161, 41 ] ok at 1
[ 163, 3 ] ok at 2
[ 163, 7 ] ok at 1
[ 163, 19 ] ok at 1
[ 163, 163 ] ok at 1
[ 165, 3 ] ok at 1
[ 165, 5 ] ok at 1
[ 165, 83 ] ok at 1
[ 167, 3 ] ok at 1
[ 167, 167 ] ok at 1
[ 169, 7 ] ok at 6
[ 169, 29 ] ok at 4
[ 169, 43 ] ok at 3
[ 171, 3 ] ok at 1
[ 171, 11 ] ok at 1
[ 173, 3 ] ok at 1
[ 173, 5 ] ok at 1
[ 173, 173 ] ok at 1
[ 175, 3 ] ok at 2
[ 175, 7 ] ok at 1
[ 175, 59 ] ok at 2
[ 177, 5 ] ok at 1
[ 177, 17 ] ok at 1
[ 177, 23 ] ok at 3
[ 177, 89 ] ok at 1
[ 179, 3 ] ok at 1
[ 179, 179 ] ok at 1
[ 181, 3 ] ok at 2
[ 181, 5 ] ok at 4
[ 181, 7 ] ok at 1
[ 181, 11 ] ok at 4
[ 181, 13 ] ok at 1
[ 181, 19 ] ok at 7
[ 181, 31 ] ok at 1
[ 181, 37 ] ok at 1
[ 181, 61 ] ok at 1
[ 181, 181 ] ok at 1
[ 185, 5 ] ok at 1
[ 185, 47 ] ok at 3
[ 187, 3 ] ok at 2
[ 187, 7 ] ok at 1
[ 189, 3 ] ok at 1
[ 189, 5 ] ok at 1
[ 191, 3 ] ok at 1
[ 191, 11 ] ok at 2
[ 191, 191 ] ok at 1
[ 193, 13 ] ok at 1
[ 193, 17 ] ok at 4
[ 193, 97 ] ok at 1
[ 193, 193 ] ok at 1
[ 195, 3 ] ok at 1
[ 197, 3 ] ok at 1
[ 197, 5 ] ok at 1
[ 197, 29 ] ok at 1
[ 197, 197 ] ok at 1
[ 199, 3 ] ok at 2
[ 199, 7 ] ok at 1
[ 199, 19 ] ok at 1
[ 199, 23 ] ok at 2
[ 199, 67 ] ok at 1
[ 199, 199 ] ok at 1