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
Keywords
Examples
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.
Links
- Robin Visser, Table of n, a(n) for n = 3..5000
Programs
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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
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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
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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
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