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Chapter 5: Problem 85

A prime number \(p\) such that \(2 p+1\) is also a prime number is called a Germain prime, named after the German mathematician Sophie Germain (1776-1831), who made major contributions to number theory. Determine whether or not each prime number is a Germain prime. 241

Short Answer

Expert verified
No, 241 is not a Germain prime.

Step by step solution

01

Verify if the given number is prime

241 is a prime number as it has only two distinct positive divisors: 1 and itself.
02

Apply the Germain prime condition

To check if 241 is a Germain prime, the formula \(2 p+1\) needs to be applied, with \(p\) as 241. This gives \(2 * 241 + 1 = 483\). The next step is to ascertain if 483 is also a prime number.
03

Step 3:Confirm if the result from Step 2 is a prime number

483 is not a prime number because it has more than two distinct positive divisors. For instance, it can be divided evenly by 3 and 161. Therefore, 241 is not a Germain prime.

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