A368055 Smallest prime number p such that x^n + y^n + z^n mod p does not take all values on Z/pZ.

5, 11, 7, 29, 5, 19, 11, 23, 5, 53, 29, 11, 5, 103, 7, 191, 5, 29, 23, 47, 5, 11, 53, 19, 5, 59, 7, 311, 5, 23, 103, 11, 5, 149, 191, 53, 5, 83, 7, 173, 5, 11, 47, 283, 5, 29, 11, 103, 5, 107, 7, 11, 5, 191, 59, 709, 5, 367, 311, 19, 5, 11, 7, 269, 5, 47, 11, 569, 5, 293, 149, 11

Offset: 4

Author

Robin Visser, Dec 09 2023

Keywords

• nonn

Comments

If there exists some prime p > 3 such that p-1 divides n, then x^n (mod p) is either 0 or 1 for all integers x, therefore giving an upper bound of a(n) <= p.

Examples

			For n = 4, x^4 + y^4 + z^4 attains all values on Z/2Z and Z/3Z, but x^4 + y^4 + z^4  == 4 (mod 5) has no solution, so a(4) = 5.
For n = 5, x^5 + y^5 + z^5 attains all values on Z/2Z, Z/3Z, Z/5Z, and Z/7Z, but x^5 + y^5 + z^5 == 4 (mod 11) has no solution, so a(5) = 11.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 4..5226 (terms 4..500 from Robert Israel)

Crossrefs

Cf. A367689.

Programs

Maple

f:= proc(n) local p,s,t,T,S,S2,S3;
  p:= 2;
  do
    p:= nextprime(p);
    T:= {$0..p-1}:
    S:= {seq(s^n mod p,s=0..p-1)};
    if S = T then next fi;
    S2:= {seq(seq(s+t mod p, s=S),t=S)};
    if S2 = T then next fi;
    S3:= {seq(seq(s+t mod p, s=S),t=S2)}:
    if S3 <> T then return p fi
  od
end proc:
map(f, [$4..100]); # Robert Israel, Jan 26 2024

Python

from itertools import combinations_with_replacement
from sympy import nextprime
def A368055(n):
    p = 1
    while (p:=nextprime(p)):
        pset = set(q:=tuple(pow(x,n,p) for x in range(p)))
        if not all(any((k-a[0]-a[1])%p in pset for a in combinations_with_replacement(q,2)) for k in range(p)):
            return p # Chai Wah Wu, Apr 04 2024

SageMath

def a(n):
    for p in Primes():
        all_values = set()
        for x in range(p):
            for y in range(p):
                for z in range(p): all_values.add((x^n+y^n+z^n)%p)
        if len(all_values) < p: return p

Formula

a(n+k*(a(n)-1)) <= a(n). - Robert Israel, Jan 26 2024