Difference between revisions of "Prime triplet"
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An example of a prime triplet is <math>\{3,5,7\}</math>. | An example of a prime triplet is <math>\{3,5,7\}</math>. | ||
| − | <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 1 is not a prime, <math>\{3,5,7\}</math> is the only prime triplet. | + | <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. |
==See Also== | ==See Also== | ||
Latest revision as of 00:59, 17 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.
