Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q2

Background

All theorems, corollaries, and definitions listed in the book's order:

Theorems and Corollaries in Elementary Number Theory (Ch 1 - 3)

All theorems and corollaries mentioned in David M. Burton’s Elementary Number Theory are listed by following the book’s order. (7th Edition) (Currently Ch 1 - 3)

RanblogRan

I will only use theorems or facts that are proved before this question. So you will not see that I quote theorems or facts from the later chapters.

Question

(a) If $1$ is added to a product of twin primes, prove that a perfect square is always obtained.
(b) Show that the sum of twin primes $p$ and $p + 2$ is divisible by $12$, provided that $p > 3$.

Solution

(a)

Let $p$ and $p + 2$ be the twin primes. $p(p + 2) + 1 = p^{2} + 2p + 1 = (p + 1)^2$. Therefore, a perfect square is always obtained.