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In [[mathematics]], a '''semiprime''' (also called '''biprime''' or '''2-[[almost prime]]''', or '''pq number''') is a [[natural number]] that is the product of two (not necessarily distinct) [[prime number]]s. The semiprimes less than 100 are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95. {{OEIS|id=A001358}}.
{{Antennas|characteristics}}


By definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and 26.
'''Radiation resistance''' is that part of an [[antenna (electronics)|antenna]]'s feedpoint resistance that is caused by the [[radiation]] of [[Electromagnetic radiation|electromagnetic]] waves from the antenna, as opposed to loss resistance (also called ohmic resistance) which generally causes the antenna to heat up. The total of radiation resistance and loss resistance is the [[electrical resistance]] of the antenna.


==Properties==
The radiation resistance is determined by the geometry of the antenna, where loss resistance is primarily determined by the materials of which it is made. While the energy lost by ohmic resistance is converted to heat, the energy lost by radiation resistance is converted to electromagnetic radiation.
The total number of [[prime factor]]s Ω(''n'') for a semiprime ''n'' is two, by definition. A semiprime is either a [[Square number|square]] of a prime or [[Square-free integer|square-free]]. The square of any prime number is a semiprime, so the largest known semiprime will always be the square of the [[largest known prime]], unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a larger number is a semiprime without knowing the two factors.<ref>Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=Semiprime ''The Prime Glossary: semiprime''] at The [[Prime Pages]]. Retrieved on 2013-09-04.</ref> A composite <math>n</math> non-divisible by primes <math>\le \sqrt[3]{n}</math> is semiprime. Various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits.<ref>{{cite web|last=Broadhurst|first=David|url=http://physics.open.ac.uk/~dbroadhu/cert/semgpch.gp|title=To prove that N is a semiprime|date=12 March 2005|accessdate=2013-09-04}}</ref> These are considered novelties, since their construction method might prove vulnerable to factorization, and because it is simpler to multiply two primes together.


For a semiprime ''n''&nbsp;=&nbsp;''pq'' the value of [[Euler's totient function]] (the number of positive integers less than or equal to ''n'' that are [[relatively prime]] to ''n'') is particularly simple when ''p'' and ''q'' are distinct:
Radiation resistance is caused by the [[radiation reaction]] of the conduction electrons in the antenna.
: &phi;(''n'') = (''p'' &minus; 1) (''q'' &minus; 1) = ''p'' ''q'' &minus; (''p'' + ''q'') + 1 = ''n'' &minus; (''p'' + ''q'') + 1.
If otherwise ''p'' and ''q'' are the same,
: &phi;(''n'') = &phi;(''p''<sup>2</sup>) = (''p'' &minus; 1) ''p'' = ''p''<sup>2</sup> &minus; ''p'' = ''n'' &minus; ''p''.


The concept of the [[prime zeta function]] can be adapted to semiprimes, which defines constants like
When electrons are accelerated, as occurs when an AC electrical field is impressed on an antenna, they will radiate electromagnetic waves. These waves carry energy that is taken from the electrons. The loss of energy of the electrons appears as an effective resistance to the movement of the electrons, analogous to the ohmic resistance caused by scattering of the electrons in the crystal lattice of the metallic conductor.
: <math>\sum_{\Omega(n)=2} \frac{1}{n^2} \approx 0.1407604</math> {{OEIS|A117543}}
: <math>\sum_{\Omega(n)=2} \frac{1}{n(n-1)} \approx 0.17105</math> {{OEIS|A152447}}
: <math>\sum_{\Omega(n)=2} \frac{\ln n}{n^2} \approx 0.28360</math> {{OEIS|A154928}}


==Applications==
[[Power (physics)|Power]] is calculated as


Semiprimes are highly useful in the area of [[cryptography]] and [[number theory]], most notably in [[public key cryptography]], where they are used by [[RSA (algorithm)|RSA]] and [[pseudorandom number generator]]s such as [[Blum Blum Shub]]. These methods rely on the fact that finding two large primes and multiplying them together (resulting in a semiprime) is computationally simple, whereas [[integer factorization|finding the original factors]] appears to be difficult. In the [[RSA Factoring Challenge]], [[RSA Security]] offered prizes for the factoring of specific large semiprimes and several prizes were awarded. The most recent such challenge closed in 2007.<ref>[http://www.rsa.com/rsalabs/node.asp?id=2092 Information Security, Governance, Risk, and Compliance - EMC]. RSA. Retrieved on 2014-05-11.</ref> <!-- The original RSA Factoring Challenge, issued in 1991, was replaced in 2001 by the New RSA Factoring Challenge; it was the latter challenge that was withdrawn in 2007. -->
:<math>P = I^2R \,</math>


In practical cryptography, it is not sufficient to choose just any semiprime; a good number must evade a number of [[Integer factorization#Special-purpose|well-known special-purpose algorithms]] that can factor numbers of certain form. The factors ''p'' and ''q'' of ''n'' should both be very large, around the same order of magnitude as the square root of ''n''; this makes [[trial division]] and [[Pollard's rho algorithm]] impractical. At the same time they should not be too close together, or else the number can be quickly factored by [[Fermat's factorization method]]. The number may also be chosen so that none of ''p''&nbsp;&minus;&nbsp;1, ''p''&nbsp;+&nbsp;1, ''q''&nbsp;&minus;&nbsp;1, or ''q''&nbsp;+&nbsp;1 are [[smooth number]]s, protecting against [[Pollard's p - 1 algorithm|Pollard's ''p''&nbsp;&minus;&nbsp;1 algorithm]] or [[Williams' p + 1 algorithm|Williams' ''p''&nbsp;+&nbsp;1 algorithm]].  However, these checks cannot take future algorithms or secret algorithms into account, introducing the possibility that numbers in use today may be broken by special-purpose algorithms.
where <math>I</math> is the [[electric current]] flowing into the feeds of the antenna and <math>P</math> is the power in the resulting electromagnetic field. The result is a virtual, effective [[Electrical resistance|resistance]]:


In 1974 the [[Arecibo message]] was sent with a radio signal aimed at a [[star cluster]]. It consisted of 1679 binary digits intended to be interpreted as a 23&times;73 [[bitmap]] image. The number 1679 = 23&times;73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns.
:<math>R = \frac{P}{I^2} \,</math>
 
This effective resistance is called the '''radiation''' resistance.


==See also==
==See also==
*[[Chen's theorem]]
*[[Abraham–Lorentz force]]
*[[Impedance of free space]]
*[[Bremsstrahlung]]


==References==
[[Category:Antennas (radio)]]
<references/>
 
== External links ==
* {{MathWorld|title=Semiprime|urlname=Semiprime}}
 
{{Divisor classes}}
{{Prime number classes}}
{{Classes of natural numbers}}
 
[[Category:Integer sequences]]
[[Category:Prime numbers]]
[[Category:Theory of cryptography]]

Revision as of 12:19, 11 August 2014



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Radiation resistance is that part of an antenna's feedpoint resistance that is caused by the radiation of electromagnetic waves from the antenna, as opposed to loss resistance (also called ohmic resistance) which generally causes the antenna to heat up. The total of radiation resistance and loss resistance is the electrical resistance of the antenna.

The radiation resistance is determined by the geometry of the antenna, where loss resistance is primarily determined by the materials of which it is made. While the energy lost by ohmic resistance is converted to heat, the energy lost by radiation resistance is converted to electromagnetic radiation.

Radiation resistance is caused by the radiation reaction of the conduction electrons in the antenna.

When electrons are accelerated, as occurs when an AC electrical field is impressed on an antenna, they will radiate electromagnetic waves. These waves carry energy that is taken from the electrons. The loss of energy of the electrons appears as an effective resistance to the movement of the electrons, analogous to the ohmic resistance caused by scattering of the electrons in the crystal lattice of the metallic conductor.

Power is calculated as

P=I2R

where I is the electric current flowing into the feeds of the antenna and P is the power in the resulting electromagnetic field. The result is a virtual, effective resistance:

R=PI2

This effective resistance is called the radiation resistance.

See also

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