# Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q8

My Solution for "Given a positive integer $n$, it can be shown that there exists an even integer $a$ that is representable as the sum of two odd primes in $n$ different ways. Confirm that the integers $60$, $78$, and $84$ can be written as the sum of two primes in 6, 7, and 8 ways, respectively."

## Background

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

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

Given a positive integer $n$, it can be shown that there exists an even integer $a$ that is representable as the sum of two odd primes in $n$ different ways. Confirm that the integers $60$, $78$, and $84$ can be written as the sum of two primes in six, seven, and eight ways, respectively.

## Solution

For $60$:

$$\begin{equation} \begin{split} 60 &= 7 + 53\\ &= 13 + 47\\ &= 17 + 43\\ &= 19 + 41\\ &= 23 + 37\\ &= 29 + 31\\ \end{split} \nonumber \end{equation}$$

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