A367689 Smallest prime number p such that x^n + y^n mod p does not take all values on Z/pZ.

7, 5, 11, 7, 29, 5, 7, 11, 23, 5, 53, 29, 7, 5, 103, 7, 191, 5, 7, 23, 47, 5, 11, 53, 7, 5, 59, 7, 311, 5, 7, 103, 11, 5, 149, 191, 7, 5, 83, 7, 173, 5, 7, 47, 283, 5, 29, 11, 7, 5, 107, 7, 11, 5, 7, 59, 709, 5, 367, 311, 7, 5, 11, 7, 269, 5, 7, 11, 569, 5, 293, 149, 7, 5, 23, 7, 317, 5

Offset

3,1

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.

Links

Robin Visser, Table of n, a(n) for n = 3..5000

Example

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

Prog

(Sage)
def a(n):
    for p in Primes():
        all_values = set()
        for x in range(p):
            for y in range(p):
                all_values.add((x^n+y^n)%p)
        if len(all_values) < p: return p
(PARI) a(n) = my(p=2); while (#setbinop((x, y)->Mod(x, p)^n+Mod(y, p)^n, [0..p-1]) == p, p=nextprime(p+1)); p; \\ Michel Marcus, Nov 27 2023
(Python)
from itertools import combinations_with_replacement
from sympy import sieve
def A367689(n):
    for p in sieve.primerange(n**4+1):
        s = set()
        for k in combinations_with_replacement({pow(x, n, p) for x in range(p)}, 2):
            s.add(sum(k)%p)
            if len(s) == p:
                break
        else:
            return p # Chai Wah Wu, Nov 27 2023

Crossrefs

Cf. A355920, A367688.

Sequence in context: A304136 A305042 A247870 * A141391 A135766 A067745

Adjacent sequences: A367686 A367687 A367688 * A367690 A367691 A367692

Keyword

nonn

Author

Robin Visser, Nov 26 2023

Status

approved