Black hole thermodynamics: Difference between revisions

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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 first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... {{OEIS|id=A001358}}.
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you have any sort of questions pertaining to where and ways to utilize [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist], you could contact us at our site.
 
Semiprime numbers also never have any composite factors other than themselves. For example, the number 26 is semiprime because its only factors are 1, 2, 13, and 26. In fact, there are no numbers that are the product of two primes that have any composite factors.
 
==Properties==
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 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:
: &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
: <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==
 
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</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. -->
 
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.
 
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.
 
==See also==
*[[Chen's theorem]]
 
==References==
<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]]

Latest revision as of 14:20, 3 January 2015

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

If you have any sort of questions pertaining to where and ways to utilize Washington DC Dentist, you could contact us at our site.

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