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

%I A367689 #22 Nov 28 2023 11:00:40
%S A367689 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,
%T A367689 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,
%U A367689 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
%N A367689 Smallest prime number p such that x^n + y^n mod p does not take all values on Z/pZ.
%C A367689 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.
%H A367689 Robin Visser, <a href="/A367689/b367689.txt">Table of n, a(n) for n = 3..5000</a>
%e A367689 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.
%e A367689 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.
%o A367689 (Sage)
%o A367689 def a(n):
%o A367689     for p in Primes():
%o A367689         all_values = set()
%o A367689         for x in range(p):
%o A367689             for y in range(p):
%o A367689                 all_values.add((x^n+y^n)%p)
%o A367689         if len(all_values) < p: return p
%o A367689 (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
%o A367689 (Python)
%o A367689 from itertools import combinations_with_replacement
%o A367689 from sympy import sieve
%o A367689 def A367689(n):
%o A367689     for p in sieve.primerange(n**4+1):
%o A367689         s = set()
%o A367689         for k in combinations_with_replacement({pow(x,n,p) for x in range(p)},2):
%o A367689             s.add(sum(k)%p)
%o A367689             if len(s) == p:
%o A367689                 break
%o A367689         else:
%o A367689             return p # _Chai Wah Wu_, Nov 27 2023
%Y A367689 Cf. A355920, A367688.
%K A367689 nonn
%O A367689 3,1
%A A367689 _Robin Visser_, Nov 26 2023