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fermat’s little theorem and euler’s generalization of fermats theorem
1. supose p is prime and that α has order 3 modulo p. what is the order of α+1? (you need to solve this for all pais (α,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 Φ(n) divides 25 n
Expert Answer
- Fermat’s Little Theorem.
- Let p be a prime which does not divide the integer a, then ap-1 = 1 (mod p).
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 Euler first published a proof in 1736, but Leibniz left virtually the same proof in an unpublished manuscript from sometime before 1683.
- Proof.
- Start by listing the first p-1 positive multiples of a:a, 2a, 3a, … (p -1)a
Suppose that ra and sa are the same modulo p, then we have r = s (mod p), so the p-1 multiples of a above are distinct and nonzero; that is, they must be congruent to 1, 2, 3, …, p-1 in some order. Multiply all these congruences together and we finda.2a.3a.….(p-1)a = 1.2.3.….(p-1) (mod p)
or better, a(p-1)(p-1)! = (p-1)! (mod p). Divide both side by (p-1)! to complete the proof.
