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

My Solution for "If pn denotes the nth prime number, put dn=pn+1pn. An open question is whether the equation dn=dn+1 has infinitely many solutions. Give five solutions."

Ran
Ran


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)

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 pn denotes the nth prime number, put dn=pn+1pn. An open question is whether the equation dn=dn+1 has infinitely many solutions. Give five solutions.

Solution

(I wrote a program for this question.)

d2=p3p2=53=2
d3=p4p3=75=2
So, d2=d3.

d15=p16p15=5347=6
d16=p17p16=5953=6
So, d15=d16

d36=p37p36=157151=6
d37=p38p37=163157=6
So, d36=d37

d39=p40p39=173167=6
d40=p41p40=179173=6
So, d39=d40

d46=p47p46=211199=12
d47=p48p47=223211=12
So, d46=d47


Read More: All My Solutions for This Book

< Chapter 3.2, Q10 Chapter 3.2, Q12 >

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Ran

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