fermat\'s little theorem and euler\'s generalization of fermats theorem 1. supos

Question

fermat's little theorem and euler's generalization of fermats theorem

1. supose p is prime and that has order 3 modulo p. what is the order of +1? (you need to solve this for all pais (,p) that satisfy the conditions of the problem.

2. Suppose p is a prime and that 2 and 3 are both primitive roots(modp). Prove that 4 and 6 are both not primitive roots(modp). (you must prove this for every p such that 2 and 3 are the primitive roots(mod p)- and there are probably infinitely many such p). Is it possible that 5 is also a primitive root(mod p).

3. find with proof, all n such that (n) divides 25 n

Explanation / Answer

It is so easy to calculate ap-1 that most elementary primality tests are built using a version of Fermat's Little Theorem rather than Wilson's Theorem.

As usual Fermat did not provide a proof (this time saying "I would send you the demonstration, if I did not fear its being too long" [Burton80, p79]). Euler first published a proof in 1736, but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.

a, 2a, 3a, ... (p -1)a

a.2a.3a.....(p-1)a = 1.2.3.....(p-1) (mod p)

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.