cp's OEIS Frontend

Let f be a polynomial with rational coefficients and G be its Galois group. By the Chebotarev density theorem, f splits modulo infinitely many primes, and the density of such primes is 1/|G|.

If n == 0 or 1 (mod 3) or n = 2 then x^n + x + 1 is irreducible over the rationals, and if n == 2 (mod 3) and n > 2 then it factors into the product of a quadratic and an irreducible factor of degree n-2 (see reference to Selmer, Theorem 1).

For all n, it appears that the Galois group of x^n + x + 1 is as large as possible, i.e. of order n! for n == 0 or 1 (mod 3), and of order 2*(n-2)! for n == 2 (mod 3).

a(n) is the smallest prime p such that x^n + x + 1 has n (not necessarily distinct) roots modulo p.

For n > 3, it appears that all roots of x^n + x + 1 are distinct modulo a(n). For n = 2 and n = 3, there is a repeated root modulo a(n). The smallest primes modulo which x^2 + x + 1 and x^3 + x + 1 split with no repeated roots are 7 and 47 respectively.

x^n - x - 1 is irreducible for all n (see link to Selmer, Theorem 1), and it appears that the Galois group is always the full symmetric group S_n.

For n > 3, it appears that all roots of x^n - x - 1 are distinct modulo a(n). For n = 2 and n = 3, there is a repeated root modulo a(n). The smallest primes modulo which x^2 - x - 1 and x^3 - x - 1 split with no repeated roots are 11 and 59 respectively.

See A371553.

Define the equivalence relation ~ on pairs of nonzero rational numbers by (b, c) ~ (b', c') if there exists a nonzero rational number k such that b' = k^4*b and c' = k^5*c. Every such pair (b, c) is equivalent to a unique pair of integers (b', c') with c' > 0 and |b'| as small as possible, which we call a primitive pair. If (b, c) ~ (b', c') and x^5 + b*x + c is irreducible and solvable by radicals, then so is x^5 + b'*x + c' by making the substitution x -> x/k and multiplying by k^5. Hence, every polynomial of the form x^5 + b*x + c with b, c nonzero rationals is equivalent to one with integer coefficients and positive constant coefficient.

An irreducible polynomial of the form x^5 + b*x + c for rational b, c is solvable by radicals if and only if its Galois group is a subgroup of the Frobenius group of order 20, which happens if and only if the resolvent sextic (x - b)^4*(x^2 - 6*b*x + 25*b^2) - 3125*c^4*x has a rational root. If b and c are integers, then such a rational root x must be an integer, by the rational root theorem. Therefore, given an integer b, we can find all such integers c by solving the quadratic Diophantine equation x^2 - (6*b + 5*y)*x + 25*b^2 = 0 for x and y, which has finitely many solutions. The values of c are then a subset of the values +-(x-b)*y^(1/4)/5.

This sequence was computed by Richard Korf in "Linear-time Disk-Based Implicit Graph Search" (see links), but was not included in the paper.

a(1) = 0; a(n) is the smallest k such that MD5(k) < MD5(a(n-1)), where integer parameters to MD5 are encoded as base-10 ASCII strings.

If we assume that MD5 behaves like a random function from N to {0, ..., 2^128-1}, the expected length of this sequence is the harmonic number H(2^128) ~= 89.3.

a(33) > 10^15.

a(1) = 0; a(n) is the smallest k such that MD5(k) > MD5(a(n-1)), where integer parameters to MD5 are encoded as base-10 ASCII strings.

If a(1) were defined as 1 instead of 0, the sequence would begin 1, 2, 3, 44, ... and then continue in the same way.

If we assume that MD5 behaves like a random function from N to {0, ..., 2^128-1}, the expected length of this sequence is the harmonic number H(2^128) ~= 89.3.

a(31) > 10^15.

This sequence was originally computed by Richard Korf, but the full sequence was not included in his paper. It was later re-computed by Tomas Rokicki.