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Analyst Programmer Alfonzo Crosser from Newcastle, usually spends time with interests which include frisbee golf - frolf, property developers properties for sale in singapore singapore and collecting music albums. Ended up in recent past visiting Puerto-Princesa Subterranean River National Park.	0	1	2	3	4	5	6	7	8	9	10
1	1
3	0	1	-1
5	0	1	-1	-1	1
7	0	1	1	-1	1	-1	-1
9	0	1	1	0	1	1	0	1	1
11	0	1	-1	1	1	1	-1	-1	-1	1	-1

Jacobi symbol (m/n) for various m (along top) and n (along left side). Only 0 ≤ m < n are shown, since due to rule (2) below any other m can be reduced modulo n. Quadratic residues are highlighted in yellow — note that no entry with a Jacobi symbol of -1 is a quadratic residue, and every quadratic residue with m ≢ 0 (mod n) has a Jacobi symbol of 1, but some entries with a Jacobi symbol of 1 (in the n=9 row) are not quadratic residues. Notice also that when either n or m is a square, all values are 0 or 1.

The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography.

Definition

For any integer $a$ and any positive odd integer $n$ the Jacobi symbol is defined as the product of the Legendre symbols corresponding to the prime factors of $n$:

${\Bigg (}{\frac {a}{n}}{\Bigg )}=\left({\frac {a}{p_{1}}}\right)^{\alpha _{1}}\left({\frac {a}{p_{2}}}\right)^{\alpha _{2}}\cdots \left({\frac {a}{p_{k}}}\right)^{\alpha _{k}}{\mbox{ where }}n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}$

$\left({\tfrac {a}{p}}\right)$ represents the Legendre symbol, defined for all integers $a$ and all odd primes $p$ by

$\left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0{\mbox{ if }}a\equiv 0{\pmod {p}}\\+1{\mbox{ if }}a\not \equiv 0{\pmod {p}}{\mbox{ and for some integer }}x,\;a\equiv x^{2}{\pmod {p}}\\-1{\mbox{ if there is no such }}x.\end{cases}}$

Following the normal convention for the empty product, $\left({\tfrac {a}{1}}\right)=1.$ The Legendre and Jacobi symbols are indistinguishable exactly when the lower argument is an odd prime, in which case they have the same value.

Properties

The following facts, even the reciprocity laws, are straightforward deductions from the definition of the Jacobi symbol and the corresponding properties of the Legendre symbol.

The Jacobi symbol is defined only when the upper argument ("numerator") is an integer and the lower argument ("denominator") is a positive odd integer.

1) If $n$ is (an odd) prime, then the Jacobi symbol ${\Bigg (}{\frac {a}{n}}{\Bigg )}$ is equal to (and written the same as) the corresponding Legendre symbol.

2) If $a\equiv b{\pmod {n}}$ then ${\Bigg (}{\frac {a}{n}}{\Bigg )}=\left({\frac {b}{n}}\right)$

3) $\left({\frac {a}{n}}\right)={\begin{cases}\;\;\,0{\mbox{ if }}\gcd(a,n)\neq 1\\\pm 1{\mbox{ if }}\gcd(a,n)=1\end{cases}}$

If either the top or bottom argument is fixed, the Jacobi symbol is a completely multiplicative function in the remaining argument:

4) $\left({\frac {ab}{n}}\right)={\Bigg (}{\frac {a}{n}}{\Bigg )}\left({\frac {b}{n}}\right)$, so $\left({\frac {a^{2}}{n}}\right)=1{\textrm {\ or\ }}0$

5) $\left({\frac {a}{mn}}\right)=\left({\frac {a}{m}}\right)\left({\frac {a}{n}}\right)$, so $\left({\frac {a}{n^{2}}}\right)=1{\textrm {\ or\ }}0$

The law of quadratic reciprocity: if m and n are odd positive coprime integers, then

6) $\left({\frac {m}{n}}\right)=\left({\frac {n}{m}}\right)(-1)^{{\tfrac {m-1}{2}}{\tfrac {n-1}{2}}}={\begin{cases}\;\;\;\left({\frac {n}{m}}\right)&{\text{if }}n\equiv 1{\pmod {4}}{\text{ or }}m\equiv 1{\pmod {4}}\\-\left({\frac {n}{m}}\right)&{\text{if }}n\equiv m\equiv 3{\pmod {4}}\end{cases}}$

and its supplements

7) $\left({\frac {-1}{n}}\right)=(-1)^{\tfrac {n-1}{2}}={\begin{cases}\;\;\,1&{\text{if }}n\equiv 1{\pmod {4}}\\-1&{\text{if }}n\equiv 3{\pmod {4}}\end{cases}}$

8) $\left({\frac {2}{n}}\right)=(-1)^{\tfrac {n^{2}-1}{8}}={\begin{cases}\;\;\,1&{\text{if }}n\equiv 1,7{\pmod {8}}\\-1&{\text{if }}n\equiv 3,5{\pmod {8}}\end{cases}}$

Like the Legendre symbol,

If $\left({\frac {a}{n}}\right)=-1$ then $a$ is a quadratic nonresidue ${\pmod {n}}$

If $a$ is a quadratic residue ${\pmod {n}}$ and $a\not \equiv 0{\pmod {n}}$, then $\left({\frac {a}{n}}\right)=1$

But, unlike the Legendre symbol

If $\left({\frac {a}{n}}\right)=1$ then $a$ may or may not be a quadratic residue ${\pmod {n}}$.

This is because for a to be a residue (mod n) it has to be a residue modulo every prime that divides n, but the Jacobi symbol will equal one if for example a is a non-residue for exactly two of the primes which divide n.

Although the Jacobi symbol can't be uniformly interpreted in terms of squares and non-squares, it can be uniformly interpreted as the sign of a permutation by Zolotarev's lemma.

The Jacobi symbol $({\tfrac {a}{n}})$ is a Dirichlet character to the modulus n.

Calculating the Jacobi symbol

The above formulas lead to an efficient O((log a)(log b)) algorithm for calculating the Jacobi symbol, analogous to the Euclidean algorithm for finding the GCD of two numbers. (This should not be surprising in light of rule 3).

1. Reduce the "numerator" modulo the "denominator" using rule 2.
2. Extract any factors of 2 from the "numerator" using rule 4 and evaluate them using rule 8.
3. If the "numerator" is 1, rules 3 and 4 give a result of 1. If the "numerator" and "denominator" are not coprime, rule 3 gives a result of 0.
4. Otherwise, the "numerator" and "denominator" are now odd positive coprime integers, so we can flip the symbol using rule 6, then return to step 1.

Example of calculations

The Legendre symbol $({\tfrac {a}{p}})$ is only defined for odd primes p. It obeys the same rules as the Jacobi symbol (i.e., reciprocity and the supplementary formulas for $({\tfrac {-1}{p}})$ and $({\tfrac {2}{p}})$ and multiplicativity of the "numerator".)

Problem: Given that 9907 is prime, calculate $\left({\frac {1001}{9907}}\right).$

Using the Legendre symbol

$\left({\frac {1001}{9907}}\right)=\left({\frac {7}{9907}}\right)\left({\frac {11}{9907}}\right)\left({\frac {13}{9907}}\right).$

$\left({\frac {7}{9907}}\right)=-\left({\frac {9907}{7}}\right)=-\left({\frac {2}{7}}\right)=-1$

$\left({\frac {11}{9907}}\right)=-\left({\frac {9907}{11}}\right)=-\left({\frac {7}{11}}\right)=\left({\frac {11}{7}}\right)=\left({\frac {4}{7}}\right)=1$

$\left({\frac {13}{9907}}\right)=\left({\frac {9907}{13}}\right)=\left({\frac {1}{13}}\right)=1$

$\left({\frac {1001}{9907}}\right)=-1$

Using the Jacobi symbol

$\left({\frac {1001}{9907}}\right)=\left({\frac {9907}{1001}}\right)=\left({\frac {898}{1001}}\right)=\left({\frac {2}{1001}}\right)\left({\frac {449}{1001}}\right)=\left({\frac {449}{1001}}\right)$

$=\left({\frac {1001}{449}}\right)=\left({\frac {103}{449}}\right)=\left({\frac {449}{103}}\right)=\left({\frac {37}{103}}\right)=\left({\frac {103}{37}}\right)$

$=\left({\frac {29}{37}}\right)=\left({\frac {37}{29}}\right)=\left({\frac {8}{29}}\right)=\left({\frac {2}{29}}\right)^{3}=-1.$

The difference between the two calculations is that when the Legendre symbol is used the "numerator" has to be factored into prime powers before the symbol is flipped. This makes the calculation using the Legendre symbol significantly slower than the one using the Jacobi symbol, as there is no known polynomial-time algorithm for factoring integers. In fact, this is why Jacobi introduced the symbol.

Primality testing

There is another way the Jacobi and Legendre symbols differ. If the Euler criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol, and in fact may not even be -1 or 1. For example,

$\left({\frac {19}{45}}\right)=1\quad {\textrm {and}}\quad 19^{(45-1)/2}\equiv 1{\pmod {45}}$

$\left({\frac {8}{21}}\right)=-1\quad {\textrm {but}}\quad 8^{(21-1)/2}\equiv 1{\pmod {21}}$

$\left({\frac {5}{21}}\right)=1\quad {\textrm {but}}\quad 5^{(21-1)/2}\equiv 16{\pmod {21}}$

So if it's not known whether a number n is prime or composite, we can pick a random number a, calculate the Jacobi symbol $({\tfrac {a}{n}})$ and compare it with Euler's formula; if they differ modulo n, then n is composite; if they're the same modulo n for many different values of a, then n is "probably prime".

This is the basis for the probabilistic Solovay–Strassen primality test and refinements such as the Baillie-PSW primality test and the Miller–Rabin primality test.

See also

• The Kronecker symbol is a generalization of the Jacobi symbol to all integers.
• The power residue symbol is a generalization for third, fourth, and higher powers.

Notes

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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
  
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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

• 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

External links

• Calculate Jacobi symbol shows the steps of the calculation.