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

My solutions for Burton's Elementary Number Theory Problems 3.3 (7th Edition)

## Table of Contents

## Background

These are my solutions while I was studying number theory using this book. I will use the theorems and definitions mentioned in the book.

If you don't have the book or are unsure which theorem or definition I am using here, you can check out this article, where I listed all theorems, corollaries, and definitions by following the book's order.

If you haven't heard of this book and are interested in learning number theory, I strongly recommend it since it is quite friendly for beginners. You can check out the book by this link:

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

## All Problems in 3.2 (p. 49 - p. 50)

### Q1

Verify that the integers $1949$ and $1951$ are twin primes.

### Q2

(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 $p > 3$.

### Q3

Find all pairs of primes $p$ and $q$ satisfying $p - q = 3$.

### Q4

Sylvester ($1896$) rephrased the Goldbach conjecture: Every even integer $2n$ greater than $4$ is the sum of two primes, one larger than $n/2$ and the other less than $3n/2$. Verify this version of the conjecture for all even integers between $6$ and $76$.

### Q5

In 1752, Goldbach submitted the following conjecture to Euler: Every odd integer can be written in the form $p + 2a^2$, where $p$ is either a prime or $1$ and $a \geq 0$. Show that the integer $5777$ refutes this conjecture.

### Q6

Prove that the Goldbach conjecture that every even integer greater than $2$ is the sum of two primes is equivalent to the statement that every integer greater than $5$ is the sum of three primes.

[Hint: If $2n - 2 =p_{1}+ p_{2}$, then $2n =p_{1}+ p_{2} + 2$ and $2n + 1 =p_{1}+ p_{2} + 3$.]

### Q7

A conjecture of Lagrange ($1775$) asserts that every odd integer greater than $5$ can be written as a sum $p_{1} + 2p_{2}$, where $p_{1}$, $p_{2}$ are both primes. Confirm this for all odd integers through $75$.

### Q8

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.

### Q9

(a) For $n > 3$, show that the integers $n$, $n + 2$, $n + 4$ cannot all be prime.

(b) Three integers $p$, $p + 2$, $p + 6$, which are all prime, are called a $\textit{prime-triplet}$. Find five sets of prime-triplets.

### Q10

Establish that the sequence $$(n + 1)! - 2, (n + 1)! - 3, ... , (n + 1)! - (n + 1)$$ produces $n$ consecutive composite integers for $n > 2$.

### Q11

Find the smallest positive integer $n$ for which the function $f(n) = n^2 + n + 17$ is composite. Do the same for the functions $g(n) = n^2 + 21n + 1$ and $h(n) = 3n^2 + 3n + 23$.

### Q12

Let $p_{n}$ denote the $n$th prime number. For $n \geq 3$, prove that $p_{n+3}^{2} < p_{n}p_{n+1}p_{n+2}$.

[Hint: Note that $p_{n+3}^{2} < 4p_{n+2}^2 <8p_{n+1}p_{n+2}$.]

### Q13

Apply the same method of proof as in Theorem 3.6 to show that there are infinitely many primes of the form $6n + 5$.

### Q14

Find a prime divisor of the integer $N = 4(3 \cdot 7 \cdot 11) - 1$ of the form $4n + 3$. Do the same for $N = 4(3 \cdot 7 \cdot 11 \cdot 15) - 1$.

### Q15

Another unanswered question is whether there exists an infinite number of sets of five consecutive odd integers of which four are primes. Find five such sets of integers.

### Q16

Let the sequence of primes, with $1$ adjoined, be denoted by $p_{0} = 1, p_{1} = 2, p_{2} = 3, p_{3} = 5, ....$ For each $n \geq 1$, it is known that there exists a suitable choice of coefficients $\epsilon_{k} = \pm 1$ such that

$$ \begin{equation} \begin{split} p_{2n} = p_{2n - 1} + \sum_{k = 0}^{2n - 2} \epsilon_{k}p_{k} \qquad p_{2n + 1} = 2p_{2n} + \sum_{k = 0}^{2n - 1} \epsilon_{k}p_{k} \end{split} \nonumber \end{equation} $$To illustrate:

$$ 13 = 1 + 2 - 3 - 5 + 7 + 11 $$and

$$ 17 = 1 + 2 - 3 - 5 + 7 - 11 + 2 \cdot 13 $$Determine similar representations for the primes $23, 29, 31,$ and $37$.

### Q17

In $1848$, de Polignac claimed that every odd integer is the sum of a prime and a power of $2$. For example, $55 = 47 + 2^{3} = 23 + 2^{5}$. Show that the integers $509$ and $877$ discredit this claim.

### Q18

(a) If $p$ is a prime and $p \not \mid b$, prove that in the arithmetic progression $$a, a + b, a + 2b, a + 3b, ...$$ every $pth$ term is divisible by $p$.

[Hint: Because $gcd(p, b) = 1$, there exist integers $r$ and $s$ satisfying $pr+ bs = 1$.

Put $n_{k} = kp - as$ for $k= 1, 2, ...$ and show that $p \mid (a+ n_{k}b)$.]

(b) From part (a), conclude that if $b$ is an odd integer, then every other term in the indicated progression is even.

### Q19

In $1950$, it was proved that any integer $n > 9$ can be written as a sum of distinct odd primes. Express the integers $25$, $69$, $81$, and $125$ in this fashion.

### Q20

If $p$ and $p^{2} + 8$ are both prime numbers, prove that $p^{3} + 4$ is also prime.

### Q21

(a) For any integer $k > 0$, establish that the arithmetic progression $$a + b, a + 2b, a + 3b, ...$$ where $gcd(a, b) = 1$, contains $k$ consecutive terms that are composite.

[Hint: Put $n =(a+ b)(a + 2b) \cdots (a+ kb)$ and consider the $k$ terms $a+ (n + 1)b, a+ (n + 2)b, ... , a+ (n + k)b$.]

(b) Find five consecutive composite terms in the arithmetic progression $$6, 11, 16,21,26,31,36, ...$$

### Q22

Show that $13$ is the largest prime that can divide two successive integers of the form $n^{2} + 3$.

### Q23

(a) The arithmetic mean of the twin primes $5$ and $7$ is the triangular number $6$. Are there any other twin primes with a triangular mean?

(b) The arithmetic mean of the twin primes $3$ and $5$ is the perfect square $4$. Are there any other twin primes with a square mean?

### Q24

Determine all twin primes $p$ and $q = p + 2$ for which $pq - 2$ is also prime.

### Q25

Let $p_{n}$ denote the $n$th prime. For $n > 3$, show that $$p_{n} < p_{1} + p_{2} + \cdots + p_{n-1}$$

[Hint: Use induction and the Bertrand conjecture.]

### Q26

Verify the following:

(a) There exist infinitely many primes ending in $33$, such as $233$, $433$, $733$, $1033, ....$

[Hint: Apply Dirichlet's theorem.]

(b) There exist infinitely many primes that do not belong to any pair of twin primes.

[Hint: Consider the arithmetic progression $21k + 5$ for $k= 1, 2, ....$ ]

(c) There exists a prime ending in as many consecutive $1$'s as desired.

[Hint: To obtain a prime ending in $n$ consecutive $1$'s, consider the arithmetic progression $10^{n}k +R_{n}$ for $k= 1, 2, ....$ ]

(d) There exist infinitely many primes that contain but do not end in the block of digits $123456789$.

[Hint: Consider the arithmetic progression $10^{n}k + 1234567891$ for $k= 1, 2, ....$]

### Q27

Prove that for every $n \geq 2$ there exists a prime $p$ with $p \leq n < 2p$.

[Hint: In the case where $n = 2k + 1$, then by the Bertrand conjecture there exists a prime $p$ such that $k < p < 2k$.]

### Q28

(a) If $n > 1$, show that $n!$ is never a perfect square.

(b) Find the values of $n \geq 1$ for which $$n! + (n + 1)! + (n + 2)!$$ is a perfect square.

[Hint: Note that $n! + (n + 1)! + (n + 2)! = n!(n + 2)^{2}$ .]

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