A255867 Least m > 0 such that gcd(m^n+17, (m+1)^n+17) > 1, or 0 if there is no such m.

1, 0, 1, 1925, 1, 189812175, 1, 2, 1, 116, 1, 55508752881180794569675021, 1, 337276, 1, 230, 1, 162, 1, 2628, 1, 15, 1, 3604979675443168377172749, 1, 53, 1, 248, 1, 254, 1, 5998484614, 1, 1323, 1, 2, 1, 42750021, 1, 51, 1, 17870, 1, 108, 1, 87, 1, 8274, 1, 2, 1, 35, 1, 4049, 1, 308, 1, 8885, 1, 2805086, 1

Offset

0,4

Comments

See A118119, which is the main entry for this class of sequences.

Links

Table of n, a(n) for n=0..60.

Formula

a(2k) = 1 for k>=0, because gcd(1^(2k)+17, 2^(2k)+17) = gcd(18, 4^k-1) >= 3 since 4 = 1 (mod 3).

Example

For n=0, gcd(m^0+17, (m+1)^0+17) = gcd(18, 18) = 18, therefore a(0)=1, the smallest possible (positive) m-value.
For n=1, gcd(m^n+17, (m+1)^n+17) = gcd(m+17, m+18) = 1, therefore a(1)=0.
For n=2, see formula with k=0.
For n=3, gcd(1925^3+17, 1926^3+17) = 1951 and (m, m+1) = (1925, 1926) is the smallest pair which yields a GCD > 1 here.

Maple

f:= proc(n) local q1, q2, r, m, bestm, p, A;
  q1:= m^n + 17;
  q2:= (m+1)^n + 17;
  r:= resultant(q1, q2, m);
  bestm:= infinity;
  for p in numtheory:-factorset(r) do
    A:= [msolve(q1, p)];
    A:= select(s -> eval(q2, s) mod p = 0, A);
    bestm:= min(bestm, op(map(s -> subs(s, m), A)));
  od;
  if bestm = infinity then -1 else bestm fi
end proc:
f(0):= 1: f(1):=0:
map(f, [$1..26]); # Robert Israel, May 31 2019

Mathematica

A255867[n_] := Module[{m = 1}, While[GCD[m^n + 17, (m + 1)^n + 17] <= 1, m++]; m]; Join[{1, 0}, Table[A255867[n], {n, 2, 10}]] (* Robert Price, Oct 16 2018 *)

Prog

(PARI) a(n, c=17, L=10^7, S=1)={n!=1 && for(a=S, L, gcd(a^n+c, (a+1)^n+c)>1 && return(a))}
(Python)
from sympy import primefactors, resultant, nthroot_mod
from sympy.abc import m
def A255867(n):
    if n == 0: return 1
    k = 0
    for p in primefactors(resultant(m**n+17, (m+1)**n+17)):
        for d in (a for a in nthroot_mod(-17, n, p, all_roots=True) if pow(a+1, n, p)==-17%p):
            k = min(d, k) if k else d
    return k  # Chai Wah Wu, May 07 2024

Crossrefs

Cf. A118119, A255832, A255852-A255869

Sequence in context: A107564 A135648 A363821 * A326018 A202051 A283949

Adjacent sequences: A255864 A255865 A255866 * A255868 A255869 A255870

Keyword

nonn,hard

Author

M. F. Hasler, Mar 09 2015

Extensions

a(5)-a(22) from Hiroaki Yamanouchi, Mar 12 2015

a(23)-a(60) from Max Alekseyev, Aug 06 2015

Status

approved