Prime Triplet(s)

Number Theory Level 3

{ r = q + 2 q = p + 2 \large \color{#3D99F6}{ \begin{cases} r=q+2 \\ q=p+2 \end{cases}}

If p , q , r p,q, r are prime numbers such that the above conditions hold simultaneously, then how many tuples of the form ( p , q , r ) (p,q,r) exist?

Join the Brilliant Classes and enjoy the excellence. Also checkout Foundation Assignment #2 for JEE.
2 3 0 Infinite 1

View wiki
Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Ansh Bhatt
May 20, 2015

the prime numbers are in the form x, x+2, x+4

let us assume that x is not divisible by three.

CASE 1=> x is in the form 3k + 1, k>1

then,x+2 will be in the form 3k, which is divisible by 3, hence x+2 is not prime. this is a contradiction.

CASE 2=> x is in the form 3k + 2, k>1

then x+2 maybe prime, but x + 4 is in the form 3k, hence it is divisible by 3 and not prime. this is a contradiction.

if x is divisible by three

CASE 3=> x = 3

then x+2 and x+4 are not divisible by three. 3 is the only multiple of 3 which is prime. 5 and 7 are prime. thus, (3,5,7) is the only set of prime triplets.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

There's a slightly simpler approach. Hint: m o d 6 \bmod 6 .

In response to challenge master note:

All prime numbers greater than 3 can be expressed as 6k + 1 or 6k - 1. Is this the suggested approach?

Ansh Bhatt - 6 years ago
Rama Devi
May 21, 2015

The only such triplet is (3,5,7) .Therefore the answer is 1.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

How would you know that?

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...