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A twin prime is a prime number that has a prime gap of two, in other word, differs from another prime number by two, for example the twin prime pair (41, 43). Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes appear despite the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger due to the prime number theorem (the "average gap" between primes less than n is log(n)). Endocrinologist Grippo from Capreol, likes to spend time backgammon, property developers in singapore and creating a house. Has lately finished a trip to China Danxia.

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History

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes p such that p + 2 is also prime. In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs p and p′ such that p′ − p = 2k. The case k = 1 is the twin prime conjecture.

A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

On April 17, 2013, Yitang Zhang announced a proof that for some integer N that is at most 70 million, there are infinitely many pairs of primes that differ by N.^[1]^[2] Zhang's paper was accepted by Annals of Mathematics in early May 2013.^[3] Terence Tao subsequently proposed a Polymath project collaborative effort to optimize Zhang’s bound;^[4] as of November 28, 2013, Thomas Engelsma claims to have reduced the bound to N = 576.^[5]

Brun's theorem

In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent.^[6] This famous result, called Brun's theorem, was the first use of the Brun sieve and helped initiate the development of modern sieve theory. The modern version of Brun's argument can be used to show that the number of twin primes less than N does not exceed

\frac{C N}{(\log N)^{2}}

for some absolute constant C > 0.^[7]

Other theorems weaker than the twin-prime conjecture

In 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many primes p such that (p′ − p) < (c ln p) where p′ denotes the next prime after p. This result was successively improved; in 1986 Helmut Maier showed that a constant c < 0.25 can be used. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to c = 0.085786… In 2005, Goldston, János Pintz and Yıldırım established that c can be chosen to be arbitrarily small^[8]^[9]

{lim inf}_{n \to \infty} \frac{p_{n + 1} - p_{n}}{\log p_{n}} = 0 .

By assuming the Elliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely many n such that at least two of n, n + 2, n + 6, n + 8, n + 12, n + 18, or n + 20 are prime. Under a stronger hypothesis they showed that for infinitely many n, at least two of n, n + 2, n + 4, and n + 6 are prime.

The result of Zhang,

{lim inf}_{n \to \infty} (p_{n + 1} - p_{n}) < N w i t h N = 7 \times 1 0^{7},

is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath project optimization of Zhang’s bound and Maynard claims to have reduced the bound to N = 600 are farther improvements.

Every twin prime pair except (3, 5) is of the form (6n − 1, 6n + 1) for some natural number n, and with the exception of n = 1, n must end in 0, 2, 3, 5, 7, or 8. Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

It has been proven that the pair (m, m + 2) is a twin prime if and only if

4 ((m - 1)! + 1) \equiv - m (\mod m (m + 2)) .

If m − 4 or m + 6 is also prime then the three primes are called a prime triplet.

Largest known twin prime pair

On January 15, 2007 two distributed computing projects, Twin Prime Search and PrimeGrid found the largest known twin primes, 2003663613 · 2¹⁹⁵⁰⁰⁰ ± 1. The numbers have 58711 decimal digits. Their discoverer was Eric Vautier of France.

On August 6, 2009 those same two projects announced that a new record twin prime had been found.^[10] It is 65516468355 · 2³³³³³³ ± 1.^[11] The numbers have 100355 decimal digits.

On December 25, 2011 PrimeGrid announced that yet another record twin prime had been found.^[12] It is 3756801695685 · 2⁶⁶⁶⁶⁶⁹ ± 1.^[13] The numbers have 200700 decimal digits.

An empirical analysis of all prime pairs up to 4.35 · 10¹⁵ shows that if the number of such pairs less than x is f(x)·x/(log x)² then f(x) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity.

There are 808,675,888,577,436 twin prime pairs below 10¹⁸.^[14]

The limiting value of f(x) is conjectured to equal twice the twin prime constant (not to be confused with Brun's constant)

2 \prod_{\binom{p p r i m e}{p \geq 3}} (1 - \frac{1}{(p - 1)^{2}}) = 1.3203236 \dots;

Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. this conjecture would imply the twin prime conjecture, but remains unresolved.

The twin prime conjecture would give a better approximation, as with the prime counting function, by

π_{2} (x) \approx 2 C_{2} {li}_{2} (x) = 2 C_{2} \int_{2}^{x} \frac{d t}{{(\log_{e} t)}^{2}} .

Properties

The first few twin prime pairs are:

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), … Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

The only even prime is 2; except for the pair (2, 3), twin primes are as closely spaced as possible for two primes.

Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3. Five is therefore the only prime that is part of two pairs. Along the same lines, other than the first pair, the number centered between the twin primes must always be divisible by 6. The lower member of a pair is by definition a Chen prime.

First Hardy–Littlewood conjecture

The Hardy–Littlewood conjecture (after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. Let π₂(x) denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C₂ as^[15]

C_{2} = \prod_{p \geq 3} \frac{p (p - 2)}{(p - 1)^{2}} \approx 0.660161815846869573927812110014 \dots

Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. (here the product extends over all prime numbers p ≥ 3). Then the conjecture is that

π_{2} (n) \sim 2 C_{2} \frac{n}{(\ln n)^{2}} \sim 2 C_{2} \int_{2}^{n} \frac{d t}{(\ln t)^{2}}

in the sense that the quotient of the two expressions tends to 1 as n approaches infinity.^[16] (The second ~ is not part of the conjecture and is proved by integration by parts.)

This conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem.

Polignac's conjecture

Polignac's conjecture from 1849 states that for every positive even natural number k, there are infinitely many consecutive prime pairs p and p′ such that p′ − p = k (i.e. there are infinitely many prime gaps of size k). The case k = 2 is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of k, but Zhang's result proves that it is true for at least one (currently unknown) value of k.

Isolated prime

An isolated prime is a prime number p such that neither p − 2 nor p + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 is an isolated prime since 21 and 25 are both composite.

The first few isolated primes are

2, 23, 37, 47, 53, 67, 79, 83, 89, 97, … Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

References

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External links

Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

my web-site http://himerka.com/
Top-20 Twin Primes at Chris Caldwell's Prime Pages.
Xavier Gourdon, Pascal Sebah: Introduction to Twin Primes and Brun's Constant
"Official press release" of 58711-digit twin prime record.
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The 20 000 first twin primes
Polymath: Bounded gaps between primes
Sudden Progress on Prime Number Problem Has Mathematicians Buzzing

Template:Prime number classes

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Template:Cite web
↑ Template:Cite web
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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Bateman & Diamond (2004) p. 313
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Template:Cite web
↑ Template:Cite web
↑ Template:Cite web
↑ Template:Cite web
↑ Template:Cite web
↑ Template:Cite web
↑ Bateman & Diamond (2004) pp.334–335

[1] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[2] Template:Cite doi

[3] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[4] Template:Cite web

[5] Template:Cite web

[6] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[BD313-7] Bateman & Diamond (2004) p. 313

[8] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[9] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

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[BD3345-16] Bateman & Diamond (2004) pp.334–335

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-Violet is the title men and women use to simply call her and she fully enjoys this identify. To play basketball is one thing he would by no means give up. Managing persons has been my job for some time. Mississippi is exactly where our home is.<br><br>Take a look at my web blog: [http://Apuestasdeportivasenlinea.weebly.com/apuestas-online.html ir Ahora]
+A '''twin prime''' is a [[prime number]] that has a [[prime gap]] of  [[2 (number)|two]], in other word, differs from another prime number by two, for example the twin prime pair ([[41 (number)|41]], [[43 (number)|43]]). Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is '''prime twin''' or '''prime pair'''. Twin primes appear despite the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger due to the [[prime number theorem]] (the "average gap" between primes less than ''n'' is log(''n'')).
+{{unsolved|mathematics|Are there infinitely many twin primes?}}
+==History==
+The question of whether there exist infinitely many twin primes has been one of the great open questions in [[number theory]] for many years. This is the content of the '''twin prime conjecture''', which states: ''There are infinitely many primes'' ''p'' ''such that'' ''p''&nbsp;+&nbsp;2 ''is also prime.'' In 1849 [[Alphonse de Polignac|de Polignac]] made the more [[de Polignac's conjecture|general conjecture]] that for every [[natural number]] ''k'', there are infinitely many prime pairs ''p'' and ''p''&prime; such that ''p''&prime;&nbsp;−&nbsp;''p''&nbsp;=&nbsp;2''k''. The case&nbsp;''k''&nbsp;=&nbsp;1 is the twin prime conjecture.
+A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture, postulates a distribution law for twin primes akin to the [[prime number theorem]].
+On April 17, 2013, [[Yitang Zhang]] announced a proof that for some integer ''N'' that is at most 70 million, there are infinitely many pairs of primes that differ by&nbsp;''N''.<ref>{{cite journal|last=McKee|first=Maggie|title=First proof that infinitely many prime numbers come in pairs|journal=Nature|date=14 May 2013|url=http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989|issn=0028-0836}}</ref><ref>{{cite doi|10.1038/nature.2013.12989}}</ref> Zhang's paper was accepted by ''[[Annals of Mathematics]]'' in early May 2013.<ref>{{cite journal | url = http://annals.math.princeton.edu/articles/7954 | title = Bounded gaps between primes | first = Yitang | last = Zhang | journal = Annals of Mathematics | publisher = Princeton University and the Institute for Advanced Study | accessdate =May 21, 2013}}</ref>  [[Terence Tao]] subsequently proposed a [[Polymath project]] collaborative effort to optimize Zhang’s bound;<ref>{{cite web|last=Tao|first=Terence|authorlink=Terence Tao|url=http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/|title=Polymath proposal: bounded gaps between primes|date=June 4, 2013}}</ref> as of November 28, 2013, Thomas Engelsma claims to have reduced the bound to N = 576.<ref>{{cite web|url=http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes|title=Bounded gaps between primes|publisher=Polymath|accessdate=2013-11-28}}</ref>
+==Brun's theorem==
+In 1915, [[Viggo Brun]] showed that the [[Brun's constant|sum of reciprocals of the twin primes]] was convergent.<ref>{{citation | jfm=45.0330.16 | last=Brun | first=V. | authorlink=Viggo Brun | title=Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare | language=German | journal=Arch. f. Math. og Naturv. | volume=34 | number=8 | pages=3–19 | year=1915 | issn=0365-4524 }}</ref>  This famous result, called [[Brun's theorem]], was the first use of the [[Brun sieve]] and helped initiate the development of modern [[sieve theory]].  The modern version of Brun's argument can be used to show that the number of twin primes less than ''N'' does not exceed
+:<math>\frac{CN}{(\log N)^2}</math>
+for some absolute constant ''C''&nbsp;>&nbsp;0.<ref name=BD313>Bateman & Diamond (2004) p.&nbsp;313</ref>
+==Other theorems weaker than the twin-prime conjecture==
+In 1940, [[Paul Erdős]] showed that there is a [[Mathematical constant|constant]] ''c''&nbsp;<&nbsp;1 and infinitely many primes ''p'' such that (''p''&prime;&nbsp;−&nbsp;''p'')&nbsp;<&nbsp;(''c''&nbsp;ln&nbsp;''p'') where ''p''&prime; denotes the next prime after&nbsp;''p''. This result was successively improved; in 1986 [[Helmut Maier]] showed that a constant ''c''&nbsp;<&nbsp;0.25 can be used.  In 2004 [[Daniel Goldston]] and [[Cem Yıldırım]] showed that the constant could be improved further to ''c''&nbsp;=&nbsp;0.085786…  In 2005, Goldston, [[János Pintz]] and Yıldırım established that ''c'' can be chosen to be arbitrarily small<ref>{{citation
+ | last1 = Goldston | first1 = Daniel Alan | author1-link = Daniel Goldston
+ | last2 = Motohashi | first2 = Yoichi
+ | last3 = Pintz | first3 = János | author3-link = János Pintz
+ | last4 = Yıldırım | first4 = Cem Yalçın | author4-link = Cem Yıldırım
+ | arxiv = math.NT/0505300
+ | mr = 2222213
+ | issue = 4
+ | journal = Japan Academy. Proceedings. Series A. Mathematical Sciences
+ | pages = 61–65
+ | title = Small gaps between primes exist
+ | url = http://projecteuclid.org/getRecord?id=euclid.pja/1146576181
+ | volume = 82
+ | year = 2006}}.</ref><ref>{{citation
+ | last1 = Goldston | first1 = D. A. | author1-link = Daniel Goldston
+ | last2 = Graham | first2 = S. W.
+ | last3 = Pintz | first3 = J. | author3-link = János Pintz
+ | last4 = Yıldırım | first4 = C. Y. | author4-link = Cem Yıldırım
+ | arxiv = math.NT/0506067
+ | doi = 10.1090/S0002-9947-09-04788-6
+ | mr = 2515812
+ | issue = 10
+ | journal = Transactions of the American Mathematical Society
+ | pages = 5285–5330
+ | title = Small gaps between primes or almost primes
+ | volume = 361
+ | year = 2009}}.</ref>
+:<math>\liminf_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0.</math>
+By assuming the [[Elliott–Halberstam conjecture]] or a slightly weaker version, they were able to show that there are infinitely many ''n'' such that at least two of ''n'', ''n''&nbsp;+&nbsp;2, ''n''&nbsp;+&nbsp;6, ''n''&nbsp;+&nbsp;8, ''n''&nbsp;+&nbsp;12, ''n''&nbsp;+&nbsp;18, or ''n''&nbsp;+&nbsp;20 are prime.  Under a stronger hypothesis they showed that for infinitely many ''n'', at least two of&nbsp;''n'', ''n''&nbsp;+&nbsp;2, ''n''&nbsp;+&nbsp;4, and ''n''&nbsp;+&nbsp;6 are prime.
+The result of Zhang,
+:<math>\liminf_{n\to\infty}(p_{n+1}-p_n)<N\;with\;N=7\times10^7,</math>
+is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath project optimization of Zhang’s bound and Maynard claims to have reduced the bound to N = 600 are farther improvements.
+Every twin prime pair except (3,&nbsp;5) is of the form (6''n''&nbsp;&minus;&nbsp;1,&nbsp;6''n''&nbsp;+&nbsp;1) for some [[natural number]] ''n'', and with the exception of ''n''&nbsp;=&nbsp;1, ''n'' must end in 0, 2, 3, 5, 7, or&nbsp;8. {{OEIS2C|id=A002822}}
+It has been proven that the pair (''m'',&nbsp;''m''&nbsp;+&nbsp;2) is a twin prime if and only if
+:<math>4((m-1)! + 1) \equiv -m \pmod {m(m+2)}.</math>
+If ''m''&nbsp;&minus;&nbsp;4 or ''m''&nbsp;+&nbsp;6 is also prime then the three primes are called a [[prime triplet]].
+==Largest known twin prime pair==
+On January 15, 2007 two [[distributed computing]] projects, [[Twin Prime Search]] and [[PrimeGrid]] found the largest known twin primes, 2003663613 · 2<sup>195000</sup> ± 1. The numbers have 58711 [[decimal]] [[Numerical digit|digits]]. Their discoverer was [[Eric Vautier]] of [[France]].
+On August 6, 2009 those same two projects announced that a new record twin prime had been found.<ref>{{cite web | url = http://www.primegrid.com/all_news.php#188 | title = News Archive | work = [[PrimeGrid]] | accessdate = 2009-08-07 | date = 6 August 2009}}</ref> It is 65516468355 · 2<sup>333333</sup> ± 1.<ref>{{cite web | url = http://primes.utm.edu/primes/page.php?id=89650 | title = The Prime Database: 65516468355*2^333333-1 | work = [[Prime Pages]] | date = 13 August 2009 | accessdate = 2009-08-14}}</ref> The numbers have 100355 decimal digits.
+On December 25, 2011 PrimeGrid announced that yet another record twin prime had been found.<ref>{{cite web | url = http://www.primegrid.com/forum_thread.php?id=3874&nowrap=true#45670 | title = News Archive | work = [[PrimeGrid]] | accessdate = 2011-12-25 | date = 25 December 2011}}</ref> It is 3756801695685 · 2<sup>666669</sup> ± 1.<ref>{{cite web | url = http://primes.utm.edu/primes/page.php?id=103792 | title = The Prime Database: 3756801695685 · 2666669 - 1 | work = [[Prime Pages]] | date = 25 December 2011 | accessdate = 2011-12-25}}</ref> The numbers have 200700 decimal digits.
+An empirical analysis of all prime pairs up to 4.35 · 10<sup>15</sup> shows that if the number of such pairs less than <var>x</var> is f(<var>x</var>)·<var>x</var>/(log <var>x</var>)<sup>2</sup> then f(<var>x</var>) is about 1.7 for small <var>x</var> and decreases towards about 1.3 as <var>x</var> tends to infinity.
+There are 808,675,888,577,436 twin prime pairs below 10<sup>18</sup>.<ref>{{cite web | url = http://www.ieeta.pt/~tos/primes.html | title = Tables of values of pi(x) and of pi2(x) | author = Tomás Oliveira e Silva | publisher = [[Aveiro University]] | date = 7 April 2008 | accessdate = 7 January 2011}}</ref>
+The limiting value of f(<var>x</var>) is conjectured to equal twice the twin prime constant (not to be confused with [[Brun's constant]])
+:<math> 2 \prod_{\textstyle{p\;{\rm prime}\atop p \ge 3}} \left(1 - \frac{1}{(p-1)^2}\right) = 1.3203236\ldots;</math>
+{{OEIS2C|id=A114907}}
+this conjecture would imply the twin prime conjecture, but remains unresolved.
+The twin prime conjecture would give a better approximation, as with the [[prime counting function]], by
+:<math>\pi_2(x) \approx 2C_2\; \operatorname{li}_2(x) = 2C_2 \int_2^x \frac{dt}{\left(\log_e t \right)^2}.</math>
+== Properties ==
+The first few twin prime pairs are:
+:(3,  5), (5,  7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73),  (101, 103), (107, 109), (137, 139), … {{OEIS2C|id=A077800}}.
+The only even prime is 2; except for the pair (2, 3), twin primes are as closely spaced as possible for two primes.
+Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3. Five is therefore the only prime that is part of two pairs. Along the same lines, other than the first pair, the number centered between the twin primes must always be divisible by 6. The lower member of a pair is by definition a [[Chen prime]].
+==First Hardy–Littlewood conjecture==
+<!-- [[Hardy–Littlewood conjecture]] redirects here-->
+The '''Hardy–Littlewood conjecture''' (after [[G. H. Hardy]] and [[John Edensor Littlewood|John Littlewood]]) is a generalization of the twin prime conjecture.  It is concerned with the distribution of [[prime constellation]]s, including twin primes, in analogy to the [[prime number theorem]]. Let π<sub>2</sub>(''x'') denote the number of primes ''p'' ≤ ''x'' such that ''p'' + 2 is also prime. Define the '''twin prime constant''' ''C''<sub>2</sub> as<ref>{{cite web
+| url = http://oeis.org/A001692/a001692.html
+| title = A page of number theoretical constants
+| accessdate = 2011-02-02
+| year = 2007
+}}</ref>
+:<math>C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016 18158 46869 57392 78121 10014\dots</math>
+{{OEIS2C|id=A005597}}
+(here the product extends over all prime numbers ''p'' ≥ 3). Then the conjecture is that
+:<math>\pi_2(n) \sim 2 C_2 \frac{n}{(\ln n)^2} \sim 2 C_2 \int_2^n {dt \over (\ln t)^2}</math>
+in the sense that the quotient of the two expressions [[limit of a function|tends to]] 1 as ''n'' approaches infinity.<ref name=BD3345>Bateman & Diamond (2004) pp.334–335</ref>  (The second ~ is not part of the conjecture and is proved by [[integration by parts]].)
+This conjecture can be justified (but not proven) by assuming that 1 / ln ''t'' describes the [[density function]] of the prime distribution, an assumption suggested by the prime number theorem.
+==Polignac's conjecture==
+[[Polignac's conjecture]] from 1849 states that for every positive even natural number ''k'', there are infinitely many consecutive prime pairs ''p'' and ''p′'' such that ''p′ − p'' = ''k'' (i.e. there are infinitely many [[prime gap]]s of size ''k''). The case ''k'' = 2 is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of ''k'', but Zhang's result proves that it is true for at least one (currently unknown) value of k.
+==Isolated prime==
+An '''isolated prime''' is a prime number ''p'' such that neither ''p''&nbsp;&minus;&nbsp;2 nor ''p''&nbsp;+&nbsp;2 is prime. In other words, ''p'' is not part of a twin prime pair. For example, 23 is an isolated prime since 21 and 25 are both [[composite number|composite]].
+The first few isolated primes are
+:[[2 (number)|2]], [[23 (number)|23]], [[37 (number)|37]], [[47 (number)|47]], [[53 (number)|53]], [[67 (number)|67]], [[79 (number)|79]], [[83 (number)|83]], [[89 (number)|89]], [[97 (number)|97]], … {{OEIS2C|id=A007510}}.
+==See also==
+* [[Cousin prime]]
+* [[Prime gap]]
+* [[Prime k-tuple|Prime ''k''-tuple]]
+* [[Prime quadruplet]]
+* [[Prime triplet]]
+* [[Sexy prime]]
+==References==
+{{reflist|colwidth=30em}}
+* {{cite book | first1=Paul T. | last1=Bateman | author1-link=Paul T. Bateman | first2=Harold G. | last2=Diamond | title=Analytic Number Theory | publisher=World Scientific | year=2004 | isbn=981-256-080-7 | zbl=1074.11001 }}
+==Further reading==
+*{{cite book |authorlink=Neil Sloane |first=Neil |last=Sloane |authorlink2=Simon Plouffe |first2=Simon |last2=Plouffe |title=The Encyclopedia of Integer Sequences |publisher=Academic Press |location=San Diego, CA |year=1995 |isbn=0-12-558630-2 }}
+== External links ==
+* {{springer|title=Twins|id=p/t094470}}
+* [http://primes.utm.edu/top20/page.php?id=1 Top-20 Twin Primes] at Chris Caldwell's [[Prime Pages]].
+* Xavier Gourdon, Pascal Sebah: [http://numbers.computation.free.fr/Constants/Primes/twin.html ''Introduction to Twin Primes and Brun's Constant'']
+*[http://mersenneforum.org/showpost.php?p=96237&postcount=51 "Official press release"] of 58711-digit twin prime record.
+* {{MathWorld | urlname=TwinPrimes | title=Twin Primes}}
+*[http://arnflo.se/~site_files/Other/twinprimes The 20 000 first twin primes]
+*[http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes#World_records Polymath: Bounded gaps between primes]
+*[http://www.wired.com/wiredscience/2013/11/prime/ Sudden Progress on Prime Number Problem Has Mathematicians Buzzing]
+{{Prime number classes}}
+[[Category:Classes of prime numbers]]
+[[Category:Unsolved problems in mathematics]]