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

Author

M. F. Hasler, Mar 09 2015

Keywords

• nonn
• hard

Comments

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

Examples

			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.
		

Crossrefs

Cf. A118119, A255832, A255852-A255869

Programs

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 *)

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

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).

Extensions

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

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