Difference between revisions of "Prime triplet"

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== Prime Triplet ==
 
  
Three consecutive [[Prime number|prime]] numbers with a difference of two is called '''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'''.
  
Eg:- 3,5,7.
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==Discussion==
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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} mod 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.
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<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==
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*[[Twin Prime Conjecture]]
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[[Category:Number theory]]
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[[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 $\{3,5,7\}$.

$\{3,5,7\}$ 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