AoPSWiki

Looking for a challenging algebra text? Preparing for MATHCOUNTS or the AMC exams?
Check out Art of Problem Solving's Introduction to Algebra by Richard Rusczyk.

Personal tools

Search

Toolbox

Prime triplet

From AoPSWiki

A set of three prime numbers which form an arithmetic sequence with common difference two is called a prime triplet.

Discussion

An example of a prime triplet is $\{3,5,7\}$ .

$\{3,5,7\}$ turns out to be the only prime triplet. This is because any set $\{n,n+2,n+4\} \pmod 3$ becomes $\{0,2,1\}$ , $\{2,1,0\}$ , or $\{1,0,2\}$ . Therefore in every triplet there exists one number that is divisible by $3$ . The only prime number divisible by $3$ is $3$ itself, so the only triplets possible are $\{1,3,5\}$ and $\{3,5,7\}$ . Since $1$ is not a prime, $\{3,5,7\}$ is the only prime triplet.

See Also

Twin Prime Conjecture

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Prime_triplet"

Categories: Number theory | Definition

Want to learn how to tackle those tough AMC/AIME/Olympiad algebra problems? Check out Art of Problem Solving's Intermediate Algebra by Richard Rusczyk and Mathew Crawford. Over 1600 problems!

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us