Problem 8. Find all primes p such that both p − 2 and p + 2 are prime. (Make sure to show that you proved that you found all such p.) Hint: Consider three cases p = 3k, p = 3k + 1, and p = 3k + 2. Problem 8. Find all primes p such that both p - 2 and p+2 are prime. (Make sure to showthat you proved that you found all such p.) Hint: Consider three cases p= 3k, p= 3k + 1, andp=3k + 2.

Answered: Problem 8. Find all primes p such that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem 8. Find all primes p such that both p − 2 and p + 2 are prime. (Make sure to show that you proved that you found all such p.) Hint: Consider three cases p = 3k, p = 3k + 1, and p = 3k + 2.

Expert Solution

Step 1

Any prime number (in general any number) p can be written in either of the following form:

p = 3k, 3k+1 or 3k+2 [for some integer k]

[i.e., either divisible by 3,or leaves remainder 1 or 2 when divided by 3 respectively]

Now, only instance when a prime p is of the form of p = 3k is the prime number 3, as all other such forms more than 3 but are divisible by 3, hence not prime.

In this case, p-2 = 1 and p+2 = 5. All are primes.

Now, let, p = 3k+1.

In this case, p+2 = 3k+3 = 3(k+1), not prime as divisible by 3.