Problem 6. A Sophie Germain prime is a prime number p such that 2p+1 is also a prime.For example, p = 2, 3, 5 are Sophie Germain primes, but p = 7 is not (since 15 = 2.7+1 isnot a prime).Prove that if p is a Sophie Germain prime, then 2p + 1 is a divisor either of 2 - 1 or of2P+1, but not of both.

p is Sophie Germain' Prime.2p+1 is also prime.

Answered: Problem 6. A Sophie Germain prime is a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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12. Consider the following (true) claim and two possible proofs. Claim. If n is a positive integer, then n² + 3n+2 is not a prime number. Proof 1. Note that n² + 3n+2= (n + 2)(n+1). Since n 1, we have n + 1>1 and n+2 > 1, so n²+3n+2 has at least two divisors other than 1 and is therefore not prime. Proof 2. If n is even then we can write n = 2k for some positive integer k. Then n² + 3n+2 = 4k² + 6k+2 = 2(2k² +3k+1), which is even and greater than 2. Since 2 is the only even prime, it follows that n² + 3n+ 2 is not prime. Which of these proofs are valid (i.e. which of them actually prove the claim)? (a) Both proofs are valid. (b) Proof 1 is valid, but Proof 2 is not. (c) Proof 2 is valid, but Proof 1 is not. (d) Neither proof is valid.
Let p and q be prime numbers.(1) If p is not equal to q, how many generators does Zpq have?(2) If p = q, how many generators does Zp2 have
a. #5 Prove that for all integers n, it is the case that n is even if an only if 3n is even. That is, prove both implications: if n is even, then 3n is even, and if 3n is even, then n is even. b. #7 Consider the statement: for all integers a and b, if a is even and b is a multiple of 3, then ab is a multiple of 6.Then state the converse, tell if it is true, and prove or disprove.
3. Let p, q be distinct odd primes. Prove that for any positive integers m, n, the integer pq is not a perfect number.
1. Use the approach of the Great Theorem of Chapter 10 to determine if 1,679,617 is prime just as Euler did for 232 +1. (a) First find some even number a and the largest value of n such that 1, 679, 617 = a²" + 1. (b) Based on the above and Euler's Theorems what form must a prime divisor of 1,679,617 have? (c) Calculate the maximum number of possible divisors of 1,679,617 that it may be necessary to check. (d) Complete the following table for each of the possible divisors that you try. k p= (2"+1) k +1 Comment (e) Were you as lucky as Euler was? Explain.
If n and a are prime to 7 then prove that n° - a° divisible by 7?
can you amswer d,e?
Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other (both are growing larger as more sand is dumped). How fast is the base diameter increasing when the base is 18 feet long? Round your answer to three decimal places. ft/min
Theorem 12.2. For all odd prime numbers p, ((²¹)!)² = (−1)²+¹ mod p. Remark. Apply Theorem 12.1 to the expression ((²¹)!)² to see that the expression is congruent to (p-1)!(-1). Now apply Wilson's theorem to get the result.
Prove that if p > 2 is a prime, then (1 2) and (1 2... p-1 p) generate Sp.
Sophie Germain (1776-1831) studied at the École Polytechnique in Paris in a day when female students were not admitted. A Sophie Germain prime is a prime p for which 2p+ 1 is also prime. Note that the prime 11 is a Sophie Germain prime because 2.11 + 1 = 23 is also a prime; but the prime 13 is not a Sophie Germain prime because 2 13+1=27 is not a prime. For the given prime p, find 2p+1 and determine whether p is a Sophie Germain prime. p=29 For p = 29, the value of 2p + 1 is This number a prime, so p = 29 is not is a Sophie Germain prime.

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