Use primitive roots to prove that
1^100 + 2^100 + ... +(p-1)^100 = 0 (mod p)
for all except five primes p. What are these primes?
When 100/(p-1)=t in Z
then:
1^100 + 2^100 + ... +(p-1)^100 = -1(mod p)
So, the primes are:
2,3,5,11, and... 101
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Let be prime number:
if
or:
if
Proof:
Suppose so to every witch is co-prime to :
Hence,
Now, suppose , and primitive root of .
hence, congruent modulo to in some order.
Prove by yourself now that: is also system of congruent modulo .
Hence: