Difference between revisions of "Prime triplet"
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− | + | A set of three [[prime number]]s which form an arithmetic sequence with common difference two is called a '''prime triplet'''. | |
− | + | ==Discussion== | |
+ | An example of a prime triplet is <math>\{3,5,7\}</math>. | ||
− | 3,5,7 turns out to be the only prime triplet. This is because any set {n,n+2,n+4} | + | <math>\{3,5,7\}</math> turns out to be the only prime triplet. This is because any set <math>\{n,n+2,n+4\} \pmod 3</math> becomes <math>\{0,2,1\}</math>, <math>\{2,1,0\}</math>, or <math>\{1,0,2\}</math>. Therefore in every triplet there exists one number that is divisible by <math>3</math>. The only prime number divisible by <math>3</math> is <math>3</math> itself, so the only triplets possible are <math>\{1,3,5\}</math> and <math>\{3,5,7\}</math>. Since <math>1</math> is not a prime, <math>\{3,5,7\}</math> is the only prime triplet. |
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+ | ==See Also== | ||
+ | *[[Twin Prime Conjecture]] | ||
+ | |||
+ | [[Category:Number theory]] | ||
+ | [[Category:Definition]] |
Latest revision as of 23:59, 16 March 2009
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 .
turns out to be the only prime triplet. This is because any set becomes , , or . Therefore in every triplet there exists one number that is divisible by . The only prime number divisible by is itself, so the only triplets possible are and . Since is not a prime, is the only prime triplet.