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118-551: The Lucas–Lehmer test works as follows. Let M p  = 2 − 1 be the Mersenne number to test with p an odd prime . The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than M p . Define a sequence { s i } {\displaystyle \{s_{i}\}} for all i ≥ 0 by The first few terms of this sequence are 4, 14, 194, 37634, ... (sequence A003010 in

236-460: A {\displaystyle a} or p {\displaystyle p} divides b {\displaystyle b} (or both). Conversely, if a number p {\displaystyle p} has the property that when it divides a product it always divides at least one factor of the product, then p {\displaystyle p} must be prime. There are infinitely many prime numbers. Another way of saying this

354-866: A closed-form solution . Let ω = 2 + 3 {\displaystyle \omega =2+{\sqrt {3}}} and ω ¯ = 2 − 3 {\displaystyle {\bar {\omega }}=2-{\sqrt {3}}} . It then follows by induction that s i = ω 2 i + ω ¯ 2 i {\displaystyle s_{i}=\omega ^{2^{i}}+{\bar {\omega }}^{2^{i}}} for all i : and The last step uses ω ω ¯ = ( 2 + 3 ) ( 2 − 3 ) = 1. {\displaystyle \omega {\bar {\omega }}=\left(2+{\sqrt {3}}\right)\left(2-{\sqrt {3}}\right)=1.} This closed form

472-530: A bit-wise shift instruction is usually (but not always) faster than a multiply instruction and can be used to multiply (shift left) and divide (shift right) by powers of two. Multiplication by a constant and division by a constant can be implemented using a sequence of shifts and adds or subtracts. For example, there are several ways to multiply by 10 using only bit-shift and addition. In some cases such sequences of shifts and adds or subtracts will outperform hardware multipliers and especially dividers. A division by

590-555: A contradiction, suppose M p is composite, and let q be the smallest prime factor of M p . Mersenne numbers are odd, so q  > 2. Let Z q {\displaystyle \mathbb {Z} _{q}} be the integers modulo q , and let X = { a + b 3 ∣ a , b ∈ Z q } . {\displaystyle X=\left\{a+b{\sqrt {3}}\mid a,b\in \mathbb {Z} _{q}\right\}.} Multiplication in X {\displaystyle X}

708-649: A fast manner. Let x {\displaystyle x} and y {\displaystyle y} be represented as n {\displaystyle n} -digit strings in some base B {\displaystyle B} . For any positive integer m {\displaystyle m} less than n {\displaystyle n} , one can write the two given numbers as where x 0 {\displaystyle x_{0}} and y 0 {\displaystyle y_{0}} are less than B m {\displaystyle B^{m}} . The product

826-577: A flood of research into fast multiplication algorithms. This method uses three multiplications rather than four to multiply two two-digit numbers. (A variant of this can also be used to multiply complex numbers quickly.) Done recursively , this has a time complexity of O ( n log 2 ⁡ 3 ) {\displaystyle O(n^{\log _{2}3})} . Splitting numbers into more than two parts results in Toom-Cook multiplication ; for example, using three parts results in

944-984: A group, which is denoted by X * and has size | X ∗ | . {\displaystyle |X^{*}|.} One element in X that does not have an inverse is 0, so | X ∗ | ≤ | X | − 1 = q 2 − 1. {\displaystyle |X^{*}|\leq |X|-1=q^{2}-1.} Now M p ≡ 0 ( mod q ) {\displaystyle M_{p}\equiv 0{\pmod {q}}} and ω ∈ X {\displaystyle \omega \in X} , so in X . Then by equation (1), in X , and squaring both sides gives Thus ω {\displaystyle \omega } lies in X * and has inverse ω 2 p − 1 . {\displaystyle \omega ^{2^{p}-1}.} Furthermore,

1062-438: A guess by Schönhage and Strassen that this would be the optimal bound, although this remains a conjecture today. Integer multiplication algorithms can also be used to multiply polynomials by means of the method of Kronecker substitution . If a positional numeral system is used, a natural way of multiplying numbers is taught in schools as long multiplication , sometimes called grade-school multiplication , sometimes called

1180-546: A hardware multiplier. Charles Putney implemented this for the 6502 . A line of research in theoretical computer science is about the number of single-bit arithmetic operations necessary to multiply two n {\displaystyle n} -bit integers. This is known as the computational complexity of multiplication. Usual algorithms done by hand have asymptotic complexity of O ( n 2 ) {\displaystyle O(n^{2})} , but in 1960 Anatoly Karatsuba discovered that better complexity

1298-451: A natural number n {\displaystyle n} are the natural numbers that divide n {\displaystyle n} evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This leads to an equivalent definition of prime numbers: they are the numbers with exactly two positive divisors . Those two are 1 and the number itself. As 1 has only one divisor, itself, it

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1416-673: A non-zero Lucas-Lehmer residue for non-prime M p will have a different numerical value from the non-zero value calculated when s 0  = 4. It is also possible to use the starting value (2 mod  M p )(3 mod  M p ), usually denoted by 2/3 for short. This starting value equals (2 + 1) /3, the Wagstaff number with exponent p . Starting values like 4, 10, and 2/3 are universal, that is, they are valid for all (or nearly all) p . There are infinitely many additional universal starting values. However, some other starting values are only valid for

1534-537: A number of the form 2 n {\displaystyle 2^{n}} or 2 n ± 1 {\displaystyle 2^{n}\pm 1} often can be converted to such a short sequence. In addition to the standard long multiplication, there are several other methods used to perform multiplication by hand. Such algorithms may be devised for speed, ease of calculation, or educational value, particularly when computers or multiplication tables are unavailable. The grid method (or box method)

1652-493: A number, so they could not consider its primality. A few scholars in the Greek and later Roman tradition, including Nicomachus , Iamblichus , Boethius , and Cassiodorus , also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider 2 to be prime either. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed

1770-423: A prime factorization with one or more prime factors. N {\displaystyle N} is evenly divisible by each of these factors, but N {\displaystyle N} has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of N {\displaystyle N} can be in the given list. Because there is no finite list of all

1888-428: A prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, 5 2 {\displaystyle 5^{2}} denotes the square or second power of 5. {\displaystyle 5.} The central importance of prime numbers to number theory and mathematics in general stems from

2006-469: A prime. Christian Goldbach formulated Goldbach's conjecture , that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem ) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of

2124-434: A relatively simple multiplication-only stage, before these contributions are then totalled to give the final answer in a separate addition stage. The calculation 34 × 13, for example, could be computed using the grid: followed by addition to obtain 442, either in a single sum (see right), or through forming the row-by-row totals (300 + 40) + (90 + 12) = 340 + 102 = 442. This calculation approach (though not necessarily with

2242-452: A subset of all possible p , for example s 0  = 3 can be used if p  = 3 (mod 4). This starting value was often used where suitable in the era of hand computation, including by Lucas in proving M 127 prime. The first few terms of the sequence are 3, 7, 47, ... (sequence A001566 in the OEIS ). If s p −2  = 0 mod  M p then

2360-510: A sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime (the product of two primes). Also, any even integer greater than 10 can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory . Another type of problem concerns prime gaps , the differences between consecutive primes. The existence of arbitrarily large prime gaps can be seen by noting that

2478-405: A table from 1 to 200000 by Joseph Blater in 1888. Quarter square multipliers were used in analog computers to form an analog signal that was the product of two analog input signals. In this application, the sum and difference of two input voltages are formed using operational amplifiers . The square of each of these is approximated using piecewise linear circuits. Finally the difference of

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2596-484: A theorem of Wright . These assert that there are real constants A > 1 {\displaystyle A>1} and μ {\displaystyle \mu } such that are prime for any natural number n {\displaystyle n} in the first formula, and any number of exponents in the second formula. Here ⌊ ⋅ ⌋ {\displaystyle \lfloor {}\cdot {}\rfloor } represents

2714-441: A usefully explicit method to introduce the idea of multiple-digit multiplications; and, in an age when most multiplication calculations are done using a calculator or a spreadsheet, it may in practice be the only multiplication algorithm that some students will ever need. Lattice, or sieve, multiplication is algorithmically equivalent to long multiplication. It requires the preparation of a lattice (a grid drawn on paper) which guides

2832-639: A very similar algorithm to long multiplication in base 2, but modern processors have optimized circuitry for fast multiplications using more efficient algorithms, at the price of a more complex hardware realization. In base two, long multiplication is sometimes called "shift and add" , because the algorithm simplifies and just consists of shifting left (multiplying by powers of two) and adding. Most currently available microprocessors implement this or other similar algorithms (such as Booth encoding ) for various integer and floating-point sizes in hardware multipliers or in microcode . On currently available processors,

2950-506: Is Prime number A prime number (or a prime ) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number . For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1 , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of

3068-493: Is Goldbach's conjecture , which asserts that every even integer n {\displaystyle n} greater than 2 can be written as a sum of two primes. As of 2014 , this conjecture has been verified for all numbers up to n = 4 ⋅ 10 18 . {\displaystyle n=4\cdot 10^{18}.} Weaker statements than this have been proven; for example, Vinogradov's theorem says that every sufficiently large odd integer can be written as

3186-564: Is a multiple of any integer between 2 and n {\displaystyle {\sqrt {n}}} . Faster algorithms include the Miller–Rabin primality test , which is fast but has a small chance of error, and the AKS primality test , which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers . As of October 2024

3304-454: Is a prime between n {\displaystyle n} and 2 n {\displaystyle 2n} , proved in 1852 by Pafnuty Chebyshev . Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin , and

3422-417: Is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much research into the topic. The oldest and simplest method, known since antiquity as long multiplication or grade-school multiplication , consists of multiplying every digit in the first number by every digit in

3540-468: Is an introductory method for multiple-digit multiplication that is often taught to pupils at primary school or elementary school . It has been a standard part of the national primary school mathematics curriculum in England and Wales since the late 1990s. Both factors are broken up ("partitioned") into their hundreds, tens and units parts, and the products of the parts are then calculated explicitly in

3658-511: Is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. There is also a set of Diophantine equations in nine variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its positive values are prime. Other examples of prime-generating formulas come from Mills' theorem and

Lucas–Lehmer primality test - Misplaced Pages Continue

3776-424: Is called a prime number (or a prime ) if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called composite numbers . In other words, n {\displaystyle n} is prime if n {\displaystyle n} items cannot be divided up into smaller equal-size groups of more than one item, or if it

3894-537: Is common to use long multiplication with the base set to 2 , where w is the number of bits in a word, for multiplying relatively small numbers. To multiply two numbers with n digits using this method, one needs about n operations. More formally, multiplying two n -digit numbers using long multiplication requires Θ ( n ) single-digit operations (additions and multiplications). When implemented in software, long multiplication algorithms must deal with overflow during additions, which can be expensive. A typical solution

4012-459: Is conjectured to be the best possible algorithm, but lower bounds of Ω ( n log ⁡ n ) {\displaystyle \Omega (n\log n)} are not known. Karatsuba multiplication is an O( n ) ≈ O( n ) divide and conquer algorithm, that uses recursion to merge together sub calculations. By rewriting the formula, one makes it possible to do sub calculations / recursion. By doing recursion, one can solve this in

4130-479: Is defined as Clearly, this multiplication is closed, i.e. the product of numbers from X is itself in X . The size of X is denoted by | X | . {\displaystyle |X|.} Since q  > 2, it follows that ω {\displaystyle \omega } and ω ¯ {\displaystyle {\bar {\omega }}} are in X . The subset of elements in X with inverses forms

4248-402: Is denoted as and means that the ratio of π ( n ) {\displaystyle \pi (n)} to the right-hand fraction approaches 1 as n {\displaystyle n} grows to infinity. This implies that the likelihood that a randomly chosen number less than n {\displaystyle n} is prime is (approximately) inversely proportional to

4366-413: Is given by the offset logarithmic integral An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference. This difference is called the modulus of the progression. For example, is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by

4484-426: Is incomplete. The key idea is to multiply together the primes in any given list and add 1. {\displaystyle 1.} If the list consists of the primes p 1 , p 2 , … , p n , {\displaystyle p_{1},p_{2},\ldots ,p_{n},} this gives the number By the fundamental theorem, N {\displaystyle N} has

4602-417: Is known. Dirichlet's theorem on arithmetic progressions , in its basic form, asserts that linear polynomials with relatively prime integers a {\displaystyle a} and b {\displaystyle b} take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with

4720-464: Is not possible to arrange n {\displaystyle n} dots into a rectangular grid that is more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of

4838-412: Is not prime by this definition. Yet another way to express the same thing is that a number n {\displaystyle n} is prime if it is greater than one and if none of the numbers 2 , 3 , … , n − 1 {\displaystyle 2,3,\dots ,n-1} divides n {\displaystyle n} evenly. The first 25 prime numbers (all

Lucas–Lehmer primality test - Misplaced Pages Continue

4956-491: Is not prime. Again, s is set to 4 but is now updated 11−2 = 9 times: Since the final value of s is not 0, the conclusion is that M 11  = 2047 is not prime. Although M 11  = 2047 has nontrivial factors, the Lucas–Lehmer test gives no indication about what they might be. The proof of correctness for this test presented here is simpler than the original proof given by Lehmer. Recall

5074-637: Is prime. In the other direction, the goal is to show that the primality of M p {\displaystyle M_{p}} implies s p − 2 ≡ 0 ( mod M p ) {\displaystyle s_{p-2}\equiv 0{\pmod {M_{p}}}} . The following simplified proof is by Öystein J. Rödseth. Since 2 p − 1 ≡ 7 ( mod 12 ) {\displaystyle 2^{p}-1\equiv 7{\pmod {12}}} for odd p > 1 {\displaystyle p>1} , it follows from properties of

5192-399: Is sometimes denoted by P {\displaystyle \mathbf {P} } (a boldface capital P) or by P {\displaystyle \mathbb {P} } (a blackboard bold capital P). The Rhind Mathematical Papyrus , from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of

5310-408: Is that it takes more steps than long multiplication, so it can be unwieldy for large numbers. On paper, write down in one column the numbers you get when you repeatedly halve the multiplier, ignoring the remainder; in a column beside it repeatedly double the multiplicand. Cross out each row in which the last digit of the first number is even, and add the remaining numbers in the second column to obtain

5428-516: Is that the sequence of prime numbers never ends. This statement is referred to as Euclid's theorem in honor of the ancient Greek mathematician Euclid , since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by Euler , Goldbach's proof based on Fermat numbers , Furstenberg's proof using general topology , and Kummer's elegant proof. Euclid's proof shows that every finite list of primes

5546-398: Is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common). The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem , but no efficient formula for the n {\displaystyle n} -th prime

5664-458: Is to represent the number in a small base, b , such that, for example, 8 b is a representable machine integer. Several additions can then be performed before an overflow occurs. When the number becomes too large, we add part of it to the result, or we carry and map the remaining part back to a number that is less than b . This process is called normalization . Richard Brent used this approach in his Fortran package, MP. Computers initially used

5782-848: Is used in both the proof of sufficiency and the proof of necessity. The goal is to show that s p − 2 ≡ 0 ( mod M p ) {\displaystyle s_{p-2}\equiv 0{\pmod {M_{p}}}} implies that M p {\displaystyle M_{p}} is prime. What follows is a straightforward proof exploiting elementary group theory given by J. W. Bruce as related by Jason Wojciechowski. Suppose s p − 2 ≡ 0 ( mod M p ) . {\displaystyle s_{p-2}\equiv 0{\pmod {M_{p}}}.} Then for some integer k , so Multiplying by ω 2 p − 2 {\displaystyle \omega ^{2^{p-2}}} gives Thus, For

5900-472: The AKS primality test , requires Õ(n) bit operations in its best known variant and is extremely slow even for relatively small values. The Mersenne number M 3 = 2−1 = 7 is prime. The Lucas–Lehmer test verifies this as follows. Initially s is set to 4 and then is updated 3−2 = 1 time: Since the final value of s is 0, the conclusion is that M 3 is prime. On the other hand, M 11  = 2047 = 23 × 89

6018-453: The Islamic mathematician Ibn al-Haytham (Alhazen) found Wilson's theorem , characterizing the prime numbers as the numbers n {\displaystyle n} that evenly divide ( n − 1 ) ! + 1 {\displaystyle (n-1)!+1} . He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but

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6136-600: The Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test . Since 1951 all the largest known primes have been found using these tests on computers . The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects. The idea that prime numbers had few applications outside of pure mathematics

6254-520: The Meissel–Lehmer algorithm can compute exact values of π ( n ) {\displaystyle \pi (n)} faster than it would be possible to list each prime up to n {\displaystyle n} . The prime number theorem states that π ( n ) {\displaystyle \pi (n)} is asymptotic to n / log ⁡ n {\displaystyle n/\log n} , which

6372-426: The OEIS ). Then M p is prime if and only if The number s p  − 2  mod  M p is called the Lucas–Lehmer residue of p . (Some authors equivalently set s 1  = 4 and test s p −1 mod M p ). In pseudocode , the test might be written as Performing the mod M at each iteration ensures that all intermediate results are at most p bits (otherwise

6490-491: The Standard Algorithm : multiply the multiplicand by each digit of the multiplier and then add up all the properly shifted results. It requires memorization of the multiplication table for single digits. This is the usual algorithm for multiplying larger numbers by hand in base 10. A person doing long multiplication on paper will write down all the products and then add them together; an abacus -user will sum

6608-1149: The Toom-3 algorithm. Using many parts can set the exponent arbitrarily close to 1, but the constant factor also grows, making it impractical. In 1968, the Schönhage-Strassen algorithm , which makes use of a Fourier transform over a modulus , was discovered. It has a time complexity of O ( n log ⁡ n log ⁡ log ⁡ n ) {\displaystyle O(n\log n\log \log n)} . In 2007, Martin Fürer proposed an algorithm with complexity O ( n log ⁡ n 2 Θ ( log ∗ ⁡ n ) ) {\displaystyle O(n\log n2^{\Theta (\log ^{*}n)})} . In 2014, Harvey, Joris van der Hoeven , and Lecerf proposed one with complexity O ( n log ⁡ n 2 3 log ∗ ⁡ n ) {\displaystyle O(n\log n2^{3\log ^{*}n})} , thus making

6726-559: The floor function , the largest integer less than or equal to the number in question. However, these are not useful for generating primes, as the primes must be generated first in order to compute the values of A {\displaystyle A} or μ . {\displaystyle \mu .} Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them

6844-479: The fundamental theorem of arithmetic . This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly, this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes, although their ordering may differ. So, although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce

6962-431: The fundamental theorem of arithmetic : every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality . A simple but slow method of checking the primality of a given number n {\displaystyle n} , called trial division , tests whether n {\displaystyle n}

7080-442: The implicit constant explicit; this was improved to O ( n log ⁡ n 2 2 log ∗ ⁡ n ) {\displaystyle O(n\log n2^{2\log ^{*}n})} in 2018. Lastly, in 2019, Harvey and van der Hoeven came up with an galactic algorithm with complexity O ( n log ⁡ n ) {\displaystyle O(n\log n)} . This matches

7198-424: The largest known prime number is a Mersenne prime with 41,024,320 decimal digits . There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem , proven at

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7316-422: The order of ω {\displaystyle \omega } divides 2 p . {\displaystyle 2^{p}.} However ω 2 p − 1 ≠ 1 {\displaystyle \omega ^{2^{p-1}}\neq 1} , so the order does not divide 2 p − 1 . {\displaystyle 2^{p-1}.} Thus,

7434-435: The sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1. By

7552-768: The Greeks in viewing 1 as not being a number. By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and by the 17th century some of them included it as the first prime number. In the mid-18th century, Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler ; however, Euler himself did not consider 1 to be prime. Many 19th century mathematicians still considered 1 to be prime, and Derrick Norman Lehmer included 1 in his list of primes less than ten million published in 1914. Lists of primes that included 1 continued to be published as recently as 1956. However, around this time, by

7670-1131: The Legendre symbol that ( 3 | M p ) = − 1. {\displaystyle (3|M_{p})=-1.} This means that 3 is a quadratic nonresidue modulo M p . {\displaystyle M_{p}.} By Euler's criterion , this is equivalent to In contrast, 2 is a quadratic residue modulo M p {\displaystyle M_{p}} since 2 p ≡ 1 ( mod M p ) {\displaystyle 2^{p}\equiv 1{\pmod {M_{p}}}} and so 2 ≡ 2 p + 1 = ( 2 p + 1 2 ) 2 ( mod M p ) . {\displaystyle 2\equiv 2^{p+1}=\left(2^{\frac {p+1}{2}}\right)^{2}{\pmod {M_{p}}}.} This time, Euler's criterion gives Combining these two equivalence relations yields Let σ = 2 3 {\displaystyle \sigma =2{\sqrt {3}}} , and define X as before as

7788-468: The Lehmer symbols for starting values 4 and 10 when p is not 2 or 5 are related by: That is, ϵ(4,  p ) × ϵ(10,  p ) = 1 if p  = 5 or 7 (mod 8) and p ≠ 2, 5. OEIS sequence A123271 shows ϵ(4,  p ) for each Mersenne prime M p . In the algorithm as written above, there are two expensive operations during each iteration:

7906-406: The Mersenne number 2−1 is computed without using division. For example, Moreover, since s × s will never exceed M < 2, this simple technique converges in at most 1 p -bit addition (and possibly a carry from the p th bit to the 1st bit), which can be done in linear time. This algorithm has a small exceptional case. It will produce 2−1 for a multiple of the modulus rather than

8024-547: The Ottoman Empire. Napier's bones , or Napier's rods also used this method, as published by Napier in 1617, the year of his death. As shown in the example, the multiplicand and multiplier are written above and to the right of a lattice, or a sieve. It is found in Muhammad ibn Musa al-Khwarizmi 's "Arithmetic", one of Leonardo's sources mentioned by Sigler, author of "Fibonacci's Liber Abaci", 2002. The pictures on

8142-520: The biggest prime rather than growing past every x {\displaystyle x} . The growth rate of this sum is described more precisely by Mertens' second theorem . For comparison, the sum does not grow to infinity as n {\displaystyle n} goes to infinity (see the Basel problem ). In this sense, prime numbers occur more often than squares of natural numbers, although both sets are infinite. Brun's theorem states that

8260-522: The calculation and separates all the multiplications from the additions . It was introduced to Europe in 1202 in Fibonacci 's Liber Abaci . Fibonacci described the operation as mental, using his right and left hands to carry the intermediate calculations. Matrakçı Nasuh presented 6 different variants of this method in this 16th-century book, Umdet-ul Hisab. It was widely used in Enderun schools across

8378-405: The correct value of 0. However, this case is easy to detect and correct. With the modulus out of the way, the asymptotic complexity of the algorithm only depends on the multiplication algorithm used to square s at each step. The simple "grade-school" algorithm for multiplication requires O ( p ) bit-level or word-level operations to square a p -bit number. Since this happens O( p ) times,

8496-443: The deep algebraic number theory of Heegner numbers and the class number problem . The Hardy–Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values. Multiplication algorithm A multiplication algorithm

8614-404: The definition The goal is to show that M p is prime iff s p − 2 ≡ 0 ( mod M p ) . {\displaystyle s_{p-2}\equiv 0{\pmod {M_{p}}}.} The sequence ⟨ s i ⟩ {\displaystyle {\langle }s_{i}{\rangle }} is a recurrence relation with

8732-471: The development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology , such as public-key cryptography , which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra , objects that behave in a generalized way like prime numbers include prime elements and prime ideals . A natural number (1, 2, 3, 4, 5, 6, etc.)

8850-420: The difference of which is 27, which is the product of 9 and 3. In prehistoric time, quarter square multiplication involved floor function ; that some sources attribute to Babylonian mathematics (2000–1600 BC). Antoine Voisin published a table of quarter squares from 1 to 1000 in 1817 as an aid in multiplication. A larger table of quarter squares from 1 to 100000 was published by Samuel Laundy in 1856, and

8968-428: The differences among more than two prime numbers. Their infinitude and density are the subject of the first Hardy–Littlewood conjecture , which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem. Analytic number theory studies number theory through the lens of continuous functions , limits , infinite series , and

9086-461: The early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a " unit ". Writing a number as a product of prime numbers is called a prime factorization of the number. For example: The terms in the product are called prime factors . The same prime factor may occur more than once; this example has two copies of the prime factor 5. {\displaystyle 5.} When

9204-450: The early 20th century, mathematicians started to agree that 1 should not be classified as a prime number. If 1 were to be considered a prime, many statements involving primes would need to be awkwardly reworded. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with any number of copies of 1. Similarly,

9322-509: The end of the 19th century, which says that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm . Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture , that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred

9440-405: The explicit grid arrangement) is also known as the partial products algorithm . Its essence is the calculation of the simple multiplications separately, with all addition being left to the final gathering-up stage. The grid method can in principle be applied to factors of any size, although the number of sub-products becomes cumbersome as the number of digits increases. Nevertheless, it is seen as

9558-674: The first prime gap of length 8 is between the primes 89 and 97, much smaller than 8 ! = 40320. {\displaystyle 8!=40320.} It is conjectured that there are infinitely many twin primes , pairs of primes with difference 2; this is the twin prime conjecture . Polignac's conjecture states more generally that for every positive integer k , {\displaystyle k,} there are infinitely many pairs of consecutive primes that differ by 2 k . {\displaystyle 2k.} Andrica's conjecture , Brocard's conjecture , Legendre's conjecture , and Oppermann's conjecture all suggest that

9676-410: The integral part of squares divided by 4 like in the following example. Below is a lookup table of quarter squares with the remainder discarded for the digits 0 through 18; this allows for the multiplication of numbers up to 9×9 . If, for example, you wanted to multiply 9 by 3, you observe that the sum and difference are 12 and 6 respectively. Looking both those values up on the table yields 36 and 9,

9794-655: The largest gaps between primes from 1 {\displaystyle 1} to n {\displaystyle n} should be at most approximately n , {\displaystyle {\sqrt {n}},} a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at O ( ( log ⁡ n ) 2 ) . {\displaystyle O((\log n)^{2}).} Prime gaps can be generalized to prime k {\displaystyle k} -tuples , patterns in

9912-529: The left and bottom sides. Then those sums are totaled as shown. The binary method is also known as peasant multiplication, because it has been widely used by people who are classified as peasants and thus have not memorized the multiplication tables required for long multiplication. The algorithm was in use in ancient Egypt. Its main advantages are that it can be taught quickly, requires no memorization, and can be performed using tokens, such as poker chips , if paper and pencil aren't available. The disadvantage

10030-533: The modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression can have more than one prime only when its remainder a {\displaystyle a} and modulus q {\displaystyle q} are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that

10148-616: The most efficient randomized primality test for general integers, the Miller–Rabin primality test , requires O( k n log n log log n ) bit operations using FFT multiplication for an n -digit number, where k is the number of iterations and is related to the error rate. For constant k , this is in the same complexity class as the Lucas-Lehmer test. In practice however, the cost of doing many iterations and other differences leads to worse performance for Miller–Rabin. The most efficient deterministic primality test for any n -digit number,

10266-425: The multiplication s × s , and the mod M operation. The mod M operation can be made particularly efficient on standard binary computers by observing that This says that the least significant n bits of k plus the remaining bits of k are equivalent to k modulo 2−1. This equivalence can be used repeatedly until at most n bits remain. In this way, the remainder after dividing k by

10384-413: The number of bits would double each iteration). The same strategy is used in modular exponentiation . Starting values s 0 other than 4 are possible, for instance 10, 52, and others (sequence A018844 in the OEIS ). The Lucas-Lehmer residue calculated with these alternative starting values will still be zero if M p is a Mersenne prime. However, the terms of the sequence will be different and

10502-495: The number of digits in n {\displaystyle n} . It also implies that the n {\displaystyle n} th prime number is proportional to n log ⁡ n {\displaystyle n\log n} and therefore that the average size of a prime gap is proportional to log ⁡ n {\displaystyle \log n} . A more accurate estimate for π ( n ) {\displaystyle \pi (n)}

10620-498: The number of primes up to x {\displaystyle x} is asymptotic to x / log ⁡ x {\displaystyle x/\log x} , where log ⁡ x {\displaystyle \log x} is the natural logarithm of x {\displaystyle x} . A weaker consequence of this high density of primes was Bertrand's postulate , that for every n > 1 {\displaystyle n>1} there

10738-500: The order is exactly 2 p . {\displaystyle 2^{p}.} The order of an element is at most the order (size) of the group, so But q is the smallest prime factor of the composite M p {\displaystyle M_{p}} , so This yields the contradiction 2 p < 2 p − 1 {\displaystyle 2^{p}<2^{p}-1} . Therefore, M p {\displaystyle M_{p}}

10856-438: The penultimate term is s p −3  = ± 2 mod  M p . The sign of this penultimate term is called the Lehmer symbol ϵ( s 0 ,  p ). In 2000 S.Y. Gebre-Egziabher proved that for the starting value 2/3 and for p  ≠ 5 the sign is: That is, ϵ(2/3,  p ) = +1 if p  = 1 (mod 4) and p ≠ 5. The same author also proved Woltman 's conjecture that

10974-674: The prime numbers less than 100) are: No even number n {\displaystyle n} greater than 2 is prime because any such number can be expressed as the product 2 × n / 2 {\displaystyle 2\times n/2} . Therefore, every prime number other than 2 is an odd number , and is called an odd prime . Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5. The set of all primes

11092-493: The primes, there must be infinitely many primes. The numbers formed by adding one to the products of the smallest primes are called Euclid numbers . The first five of them are prime, but the sixth, is a composite number. There is no known efficient formula for primes. For example, there is no non-constant polynomial , even in several variables, that takes only prime values. However, there are numerous expressions that do encode all primes, or only primes. One possible formula

11210-427: The process of above multiplication. It keeps only one row to maintain the sum which finally becomes the result. Note that the '+=' operator is used to denote sum to existing value and store operation (akin to languages such as Java and C) for compactness. Some chips implement long multiplication, in hardware or in microcode , for various integer and floating-point word sizes. In arbitrary-precision arithmetic , it

11328-763: The product. This example uses peasant multiplication to multiply 11 by 3 to arrive at a result of 33. Describing the steps explicitly: The method works because multiplication is distributive , so: A more complicated example, using the figures from the earlier examples (23,958,233 and 5,830): This formula can in some cases be used, to make multiplication tasks easier to complete: In the case where x {\displaystyle x} and y {\displaystyle y} are integers, we have that because x + y {\displaystyle x+y} and x − y {\displaystyle x-y} are either both even or both odd. This means that and it's sufficient to (pre-)compute

11446-432: The products as soon as each one is computed. This example uses long multiplication to multiply 23,958,233 (multiplicand) by 5,830 (multiplier) and arrives at 139,676,498,390 for the result (product). In some countries such as Germany , the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier: Below pseudocode describes

11564-446: The progression contains infinitely many primes. The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes. Euler noted that the function yields prime numbers for 1 ≤ n ≤ 40 {\displaystyle 1\leq n\leq 40} , although composite numbers appear among its later values. The search for an explanation for this phenomenon led to

11682-467: The related mathematics of the infinite and infinitesimal . This area of study began with Leonhard Euler and his first major result, the solution to the Basel problem . The problem asked for the value of the infinite sum 1 + 1 4 + 1 9 + 1 16 + … , {\displaystyle 1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\dots ,} which today can be recognized as

11800-485: The result is now known as the prime number theorem . Another important 19th century result was Dirichlet's theorem on arithmetic progressions , that certain arithmetic progressions contain infinitely many primes. Many mathematicians have worked on primality tests for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877), Proth's theorem (c. 1878),

11918-409: The right show how to calculate 345 × 12 using lattice multiplication. As a more complicated example, consider the picture below displaying the computation of 23,958,233 multiplied by 5,830 (multiplier); the result is 139,676,498,390. Notice 23,958,233 is along the top of the lattice and 5,830 is along the right side. The products fill the lattice and the sum of those products (on the diagonal) are along

12036-555: The right. For 8-bit integers the table of quarter squares will have 2 −1=511 entries (one entry for the full range 0..510 of possible sums, the differences using only the first 256 entries in range 0..255) or 2 −1=511 entries (using for negative differences the technique of 2-complements and 9-bit masking, which avoids testing the sign of differences), each entry being 16-bit wide (the entry values are from (0²/4)=0 to (510²/4)=65025). The quarter square multiplier technique has benefited 8-bit systems that do not have any support for

12154-481: The ring X = { a + b 3 ∣ a , b ∈ Z M p } . {\displaystyle X=\{a+b{\sqrt {3}}\mid a,b\in \mathbb {Z} _{M_{p}}\}.} Then in the ring X , it follows that where the first equality uses the Binomial Theorem in a finite field , which is and the second equality uses Fermat's little theorem , which

12272-415: The same b {\displaystyle b} have approximately the same proportions of primes. Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that (for integer arguments) is prime infinitely often. Euler's proof that there are infinitely many primes considers

12390-546: The same result. Primes can thus be considered the "basic building blocks" of the natural numbers. Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma : If p {\displaystyle p} is a prime number and p {\displaystyle p} divides a product a b {\displaystyle ab} of integers a {\displaystyle a} and b , {\displaystyle b,} then p {\displaystyle p} divides

12508-488: The second and adding the results. This has a time complexity of O ( n 2 ) {\displaystyle O(n^{2})} , where n is the number of digits. When done by hand, this may also be reframed as grid method multiplication or lattice multiplication . In software, this may be called "shift and add" due to bitshifts and addition being the only two operations needed. In 1960, Anatoly Karatsuba discovered Karatsuba multiplication , unleashing

12626-419: The sequence n ! + 2 , n ! + 3 , … , n ! + n {\displaystyle n!+2,n!+3,\dots ,n!+n} consists of n − 1 {\displaystyle n-1} composite numbers, for any natural number n . {\displaystyle n.} However, large prime gaps occur much earlier than this argument shows. For example,

12744-620: The square root. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler ). Fermat also investigated the primality of the Fermat numbers 2 2 n + 1 {\displaystyle 2^{2^{n}}+1} , and Marin Mersenne studied the Mersenne primes , prime numbers of the form 2 p − 1 {\displaystyle 2^{p}-1} with p {\displaystyle p} itself

12862-488: The study of prime numbers come from the ancient Greek mathematicians , who called them prōtos arithmòs ( πρῶτος ἀριθμὸς ). Euclid 's Elements (c. 300 BC) proves the infinitude of primes and the fundamental theorem of arithmetic , and shows how to construct a perfect number from a Mersenne prime . Another Greek invention, the Sieve of Eratosthenes , is still used to construct lists of primes. Around 1000 AD,

12980-590: The sum of the reciprocals of twin primes , is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve the twin prime conjecture , that there exist infinitely many twin primes. The prime-counting function π ( n ) {\displaystyle \pi (n)} is defined as the number of primes not greater than n {\displaystyle n} . For example, π ( 11 ) = 5 {\displaystyle \pi (11)=5} , since there are five primes less than or equal to 11. Methods such as

13098-458: The sum of the reciprocals of the primes 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + ⋯ {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots } . At the start of the 19th century, Legendre and Gauss conjectured that as x {\displaystyle x} tends to infinity,

13216-419: The sums of reciprocals of primes, Euler showed that, for any arbitrary real number x {\displaystyle x} , there exists a prime p {\displaystyle p} for which this sum is bigger than x {\displaystyle x} . This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at

13334-747: The total time complexity is O( p ). A more efficient multiplication algorithm is the Schönhage–Strassen algorithm , which is based on the Fast Fourier transform . It only requires O( p log p log log p ) time to square a p -bit number. This reduces the complexity to O( p log p log log p ) or Õ( p ). An even more efficient multiplication algorithm, Fürer's algorithm , only needs p log ⁡ p   2 O ( log ∗ ⁡ p ) {\displaystyle p\log p\ 2^{O(\log ^{*}p)}} time to multiply two p -bit numbers. By comparison,

13452-436: The two squares is formed and scaled by a factor of one fourth using yet another operational amplifier. In 1980, Everett L. Johnson proposed using the quarter square method in a digital multiplier. To form the product of two 8-bit integers, for example, the digital device forms the sum and difference, looks both quantities up in a table of squares, takes the difference of the results, and divides by four by shifting two bits to

13570-598: The value ζ ( 2 ) {\displaystyle \zeta (2)} of the Riemann zeta function . This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis . Euler showed that ζ ( 2 ) = π 2 / 6 {\displaystyle \zeta (2)=\pi ^{2}/6} . The reciprocal of this number, 6 / π 2 {\displaystyle 6/\pi ^{2}} ,

13688-495: Was possible (with the Karatsuba algorithm ). Currently, the algorithm with the best computational complexity is a 2019 algorithm of David Harvey and Joris van der Hoeven , which uses the strategies of using number-theoretic transforms introduced with the Schönhage–Strassen algorithm to multiply integers using only O ( n log ⁡ n ) {\displaystyle O(n\log n)} operations. This

13806-742: Was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis. The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form. The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang 's 2013 proof that there exist infinitely many prime gaps of bounded size. Most early Greeks did not even consider 1 to be

13924-422: Was unable to prove it. Another Islamic mathematician, Ibn al-Banna' al-Marrakushi , observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit. Fibonacci took the innovations from Islamic mathematics to Europe. His book Liber Abaci (1202) was the first to describe trial division for testing primality, again using divisors only up to

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