Elementary Number Theory Problems 3.2 Solution (David M. Burton's 7th Edition) - Q11

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

If $p_n$ denotes the $n$th prime number, put $d_n=p_{n+1}-p_n$. An open question is whether the equation $d_n=d_{n+1}$ has infinitely many solutions. Give five solutions.

Solution

(I wrote a program for this question.)

$d_2=p_3-p_2=5-3=2$
$d_3=p_4-p_3=7-5=2$
So, $d_2=d_3$.

$d_{15}=p_{16}-p_{15}=53-47=6$
$d_{16}=p_{17}-p_{16}=59-53=6$
So, $d_{15}=d_{16}$

$d_{36}=p_{37}-p_{36}=157-151=6$
$d_{37}=p_{38}-p_{37}=163-157=6$
So, $d_{36}=d_{37}$

$d_{39}=p_{40}-p_{39}=173-167=6$
$d_{40}=p_{41}-p_{40}=179-173=6$
So, $d_{39}=d_{40}$

$d_{46}=p_{47}-p_{46}=211-199=12$
$d_{47}=p_{48}-p_{47}=223-211=12$
So, $d_{46}=d_{47}$

Read More: All My Solutions for This Book

Related Pages

< Chapter 3.2, Q10 Chapter 3.2, Q12 >