Exercise 1. We proved in the lectures that if p is prime, then Edp (d) = p. (1) Prove that if kEN and p is prime, then Edp (d) = pk. (2) Prove that if p and q are distinct primes, then Ed\pg 4 (d) = pq. (3) Prove that if m, n E N and (m, n) = 1, then (Σ∙()) (Σφ(5) - Σ ×(h). f|n h|mn (4) Prove that if n E N, then Σφ(α) = n. d|n

Q: 2 4) Prove that n + 5 > 4 n for all integers, n >0.

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Q: 7. Define p* = (-1)(p-1)/²p. If p, q are primes, then: () - = р (²).

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Q: 3. If π is a prime in Z[i], then the number of elements in Z[i], is N(π). 4. If p, q are positive…

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Q: (c) Prove that m is prime if and only if y(m) = m - 1.

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Q: prove for all ne J, 1+ 3 + 5 +..+(2n-1) = n² by induction

A: The statement is for all To prove :The above statement is true using mathematical induction.

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Q: 6. Prove that m is prime if and only if y(m) = m - 1.

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Q: "EZ¹, 2n + 3 is prime."

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Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: 11- P, X Pz X P, ... X Pn, where M= Pr. Pn are distind primes Prove by inducton, For all Ppositive…

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Q: b)+++ -1- for all n 1. ... 4 2"

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Q: 5. Prove that p(mk) = mk-1p(m) for all positive integers m and k.

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Q: Recall Fermat's Little Theorem: For any prime p and integer a, ap−1≡mod p It happens that the…

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Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

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Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

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Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Q: = 5, etc. Prove that for (7) Let P, denote the nth prime. Thus p₁ = 2, P2 = 3, P3 n≥ 5, Pn > 2n.

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Q: b) 21315η - 1= Σακ(π) κ. ΚΣΟ (n)2+(ηλι+6= Σάκη

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