Difference between revisions of "Semiprime"
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| − | In [[mathematics]], a '''semiprime''' is a [[number]] that is the [[product]] of two [[prime|primes]]. | + | In [[mathematics]], a '''semiprime''' is a [[number]] that is the [[product]] of two not necessarily distinct [[prime|primes]]. These integers are important in many contexts, including [[cryptography]]. |
==Examples== | ==Examples== | ||
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Other examples include: <math>25</math>, <math>15</math>, <math>39</math>, <math>221</math>, <math>437</math>, and <math>1537</math>. | Other examples include: <math>25</math>, <math>15</math>, <math>39</math>, <math>221</math>, <math>437</math>, and <math>1537</math>. | ||
| − | == | + | ==Examples of non-semiprimes== |
| − | + | *<math>17</math>, as it is only a prime number. | |
| − | <math>12</math> | + | *<math>12</math>, not a semiprime because it can obtained by <math>3*4</math> or <math>2*6</math>. |
| + | |||
| + | == Basic Properties== | ||
| + | Via the Sieve of Sundaram formulation of: <cmath>2n+1</cmath> being composite any time <cmath>n=2ab+a+b\quad 0<a,b<n\quad a,b,n\in\mathbb{N}</cmath>, as <math>2n+1=4ab+2a+2b+1=(2a+1)(2b+1)</math>, we can show that if and only if <math>a,b</math> are both not composite producing then <math>2n+1</math> is a semiprime. | ||
| + | |||
| + | Odd semiprimes, are able to be expressed as a difference of squares, like all other numbers that are products of numbers of same parity. | ||
| + | |||
| + | ==See Also== | ||
| + | *[[Prime]] | ||
| + | *[[Factor]] | ||
| + | *[[Prime factorization]] | ||
| + | |||
| + | [[Category:Number theory]] | ||
| + | {{stub}} | ||
Latest revision as of 19:05, 28 May 2020
In mathematics, a semiprime is a number that is the product of two not necessarily distinct primes. These integers are important in many contexts, including cryptography.
Examples
is an example of a semiprime as it is the product of two threes. 
.
is also an example as it is obtained by 
.
is an example of a semiprime as it is the product of two threes. 
.
is also an example as it is obtained by 
.Other examples include: 
, 
, 
, 
, 
, and 
.
Examples of non-semiprimes
, as it is only a prime number.
, not a semiprime because it can obtained by 
or 
.
, as it is only a prime number.
, not a semiprime because it can obtained by 
or 
.Basic Properties
Via the Sieve of Sundaram formulation of: ![\[2n+1\]](http://latex.artofproblemsolving.com/2/6/6/2665a7c0e286b0ddae96de376e2fb3919c634e11.png)
being composite any time ![\[n=2ab+a+b\quad 0<a,b<n\quad a,b,n\in\mathbb{N}\]](http://latex.artofproblemsolving.com/f/d/a/fda4cc20adcf95cee0e5353fcea3ea7738773273.png)
, as 
, we can show that if and only if 
are both not composite producing then 
is a semiprime.
Odd semiprimes, are able to be expressed as a difference of squares, like all other numbers that are products of numbers of same parity.
See Also
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