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Problem 4: Fermat’s Little Theorem. In this problem, you will give an induction proof of Fermat’s Little Theorem. You may assume the following statement, which we proved in class. For all 0, 6, 2 p âˆˆ Z with p prime, we have (a + b)^p â‰¡ a^p + b^p. Now fix a prime p âˆˆ Z and for any n âˆˆ Z, consider the statement P(n) = “n^p â‰¡ n”. Explain why P(0) and P(1) are true. If P(n) is true, prove that P(n + 1) is also true. If P(n) is true, prove that P(-n) is also true.

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02:47

4. Fermat’s Little Theorem states that if p is prime and n is not multiple of p then np-1 (mod p). Give direct proof of this fact in the cases p == 2 and p = 3. (Do not try to adapt general proofs from textbooks Just go directly by cases for n: one case if p == 2 and two cases if p = 3).

04:23

Prove or disprove that $p_{1} p_{2} \cdots p_{n}+1$ is prime for every positive integer $n,$ where $p_{1}, p_{2}, \ldots, p_{n}$ are the $n$ smallest prime numbers.

02:15

For n postive integer and p a prime, prove that p divides n if and only if p divides n?

Transcript

Hi in the given problem we have 0 power. P is equal to 0, that is equivalent to 0 p 0, equivalent 0 p and 1 to the power. P is equivalent to 1 equivalent to 1 p. So that’s the answer for the first case on the second part we have n plus 1 to the poet, is equivalent to p and p plus 1 to the power p, and that is equivalent to p n plus, 1 p, n plus 1. So, as b n is 2, so this is because p n is true, and the second third case c we have minus n to…

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