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In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A Horn, of The University of Utah, who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conjecture on the density of twin primes or their conjecture on primes of the form n² + 1; it is also a strengthening of Schinzel's hypothesis H. It remains unsolved as of January 2014.

Definition

The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of m distinct irreducible polynomials ƒ₁, ..., ƒ_m with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number p that divides their product f(n) for every positive integer n. For, if not, then one of the values of the polynomials must be equal to p, which can only happen for finitely many values of n.

An integer n is prime-generating for the given system of polynomials if every polynomial ƒ_i(n) produces a prime number when given n as its argument. If P(x) is the fraction of prime-generating integers among the positive integers less than x, then the Bateman–Horn conjecture states that

$P(x)\sim \frac{C}{D}\underset{2}{\overset{x}{\int }}\frac{dt}{(\log t{)}^m},$

where D is the product of the degrees of the polynomials and where C is the product over primes p

$C=\prod _p\frac{1-N(p)/p}{(1-1/p{)}^m}$

with $N(p)$ the number of solutions to

$f(n)\equiv 0(\mathrm{mod}p).$

Bunyakovsky's property implies $N(p)<p$ for all primes p, so each factor in the infinite product C is positive. Intuitively one then naturally expects that the constant C is itself positive, and with some work this can be proved. (Work is needed since some infinite products of positive numbers equal zero.)

Negative numbers

As stated above, the conjecture is not true: the single polynomial ƒ₁(x) = −x produces only negative numbers when given a positive argument, so the fraction of prime numbers among its values is always zero. There are two equally valid ways of refining the conjecture to avoid this difficulty:

• One may require all the polynomials to have positive leading coefficients, so that only a constant number of their values can be negative.
• Alternatively, one may allow negative leading coefficients but count a negative number as being prime when its absolute value is prime.

It is reasonable to allow negative numbers to count as primes as a step towards formulating more general conjectures that apply to other systems of numbers than the integers, but at the same time it is easy to just negate the polynomials if necessary to reduce to the case where the leading coefficients are positive.

Examples

If the system of polynomials consists of the single polynomial ƒ₁(x) = x, then the values n for which ƒ₁(n) is prime are themselves the prime numbers, and the conjecture becomes a restatement of the prime number theorem.

If the system of polynomials consists of the two polynomials ƒ₁(x) = x and ƒ₂(x) = x + 2, then the values of n for which both ƒ₁(n) and ƒ₂(n) are prime are just the smaller of the two primes in every pair of twin primes. In this case, the Bateman–Horn conjecture reduces to the Hardy–Littlewood conjecture on the density of twin primes, according to which the number of twin prime pairs less than x is

${\pi }_2(x)\sim 2\prod _{p\ge 3}\frac{p(p-2)}{(p-1{)}^2}\frac{x}{(\log x{)}^2}\approx 1.32\frac{x}{(\log x{)}^2}.$

Analogue for polynomials over a finite field

When the integers are replaced by the polynomial ring F[u] for a finite field F, one can ask how often a finite set of polynomials f_i(x) in F[u][x] simultaneously takes irreducible values in F[u] when we substitute for x elements of F[u]. Well-known analogies between integers and F[u] suggest an analogue of the Bateman–Horn conjecture over F[u], but the analogue is wrong. For example, data suggest that the polynomial

$x^3+u$

in F₃[u][x] takes (asymptotically) the expected number of irreducible values when x runs over polynomials in F₃[u] of odd degree, but it appears to take (asymptotically) twice as many irreducible values as expected when x runs over polynomials of degree that is 2 mod 4, while it (provably) takes no irreducible values at all when x runs over nonconstant polynomials with degree that is a multiple of 4. An analogue of the Bateman–Horn conjecture over F[u] which fits numerical data uses an additional factor in the asymptotics which depends on the value of d mod 4, where d is the degree of the polynomials in F[u] over which x is sampled.

References

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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
  
  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
• 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.