22 ( twenty-two ) is the natural number following 21 and preceding 23 .
33-538: 22 is a semiprime , a Smith number , and an Erdős–Woods number . 22 7 = 3.14 28 … {\displaystyle {\frac {22}{7}}=3.14{\color {red}28}\ldots } is a commonly used approximation of the irrational number π , the ratio of the circumference of a circle to its diameter . 22 can read as "two twos", which is the only fixed point of John Conway's look-and-say function . The number 22 appears prominently within sporadic groups . The Mathieu group M 22
66-422: A 23 × 73 {\displaystyle 23\times 73} bitmap image. The number 1679 = 23 ⋅ 73 {\displaystyle 1679=23\cdot 73} was chosen because it is a semiprime and therefore can be arranged into a rectangular image in only two distinct ways (23 rows and 73 columns, or 73 rows and 23 columns). Prime-counting function In mathematics ,
99-986: A log x ) as x → ∞ {\displaystyle \pi (x)=\operatorname {li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty } for some positive constant a . Here, O (...) is the big O notation . More precise estimates of π ( x ) are now known. For example, in 2002, Kevin Ford proved that π ( x ) = li ( x ) + O ( x exp ( − 0.2098 ( log x ) 3 / 5 ( log log x ) − 1 / 5 ) ) . {\displaystyle \pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-0.2098(\log x)^{3/5}(\log \log x)^{-1/5}\right)\right).} Mossinghoff and Trudgian proved an explicit upper bound for
132-613: A prime. Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately x log x {\displaystyle {\frac {x}{\log x}}} where log is the natural logarithm , in the sense that lim x → ∞ π ( x ) x / log x = 1. {\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\log x}}=1.} This statement
165-443: Is (where ⌊ x ⌋ denotes the floor function ). This number is therefore equal to when the numbers p 1 , p 2 ,…, p n are the prime numbers less than or equal to the square root of x . In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating π ( x ) : Let p 1 , p 2 ,…, p n be the first n primes and denote by Φ( m , n )
198-537: Is Riemann's R-function and μ ( n ) is the Möbius function . The latter series for it is known as Gram series. Because log x < x for all x > 0 , this series converges for all positive x by comparison with the series for e . The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as ρ log x and not log x . Folkmar Bornemann proved, when assuming
231-459: Is an important part of the application of semiprimes in the RSA cryptosystem . For a square semiprime n = p 2 {\displaystyle n=p^{2}} , the formula is again simple: φ ( n ) = p ( p − 1 ) = n − p . {\displaystyle \varphi (n)=p(p-1)=n-p.} Semiprimes are highly useful in
264-678: Is equal to π 0 ( x ) = R ( x ) − ∑ ρ R ( x ρ ) , {\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho }),} where R ( x ) = ∑ n = 1 ∞ μ ( n ) n li ( x 1 / n ) , {\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} \left(x^{1/n}\right),} μ ( n )
297-472: Is greater than π ( x ) . However, π ( x ) − li( x ) is known to change sign infinitely many times. For a discussion of this, see Skewes' number . For x > 1 let π 0 ( x ) = π ( x ) − 1 / 2 when x is a prime number, and π 0 ( x ) = π ( x ) otherwise. Bernhard Riemann , in his work On the Number of Primes Less Than a Given Magnitude , proved that π 0 ( x )
330-400: Is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to x and then to count them. A more elaborate way of finding π ( x ) is due to Legendre (using the inclusion–exclusion principle ): given x , if p 1 , p 2 ,…, p n are distinct prime numbers, then the number of integers less than or equal to x which are divisible by no p i
363-407: Is one of 26 sporadic finite simple groups , defined as the 3-transitive permutation representation on 22 points. There are also 22 regular complex apeirohedra . Twenty-two may also refer to: Semiprime In mathematics , a semiprime is a natural number that is the product of exactly two prime numbers . The two primes in the product may equal each other, so the semiprimes include
SECTION 10
#1732773365619396-1226: Is the Möbius function , li( x ) is the logarithmic integral function , ρ indexes every zero of the Riemann zeta function, and li( x ) is not evaluated with a branch cut but instead considered as Ei( ρ / n log x ) where Ei( x ) is the exponential integral . If the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then π 0 ( x ) may be approximated by π 0 ( x ) ≈ R ( x ) − ∑ ρ R ( x ρ ) − 1 log x + 1 π arctan π log x . {\displaystyle \pi _{0}(x)\approx \operatorname {R} (x)-\sum _{\rho }\operatorname {R} \left(x^{\rho }\right)-{\frac {1}{\log x}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\log x}}.} The Riemann hypothesis suggests that every such non-trivial zero lies along Re( s ) = 1 / 2 . The table shows how
429-497: Is the prime number theorem . An equivalent statement is lim x → ∞ π ( x ) li ( x ) = 1 {\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{\operatorname {li} (x)}}=1} where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of
462-492: Is the prime-counting function and p k {\displaystyle p_{k}} denotes the k th prime. Semiprime numbers have no composite numbers as factors other than themselves. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and 26, of which only 26 is composite. For a squarefree semiprime n = p q {\displaystyle n=pq} (with p ≠ q {\displaystyle p\neq q} )
495-478: Is the region of interest. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last subtrahend in the formula. For Π 0 ( x ) we have a more complicated formula Again, the formula is valid for x > 1 , while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value. The first term li( x )
528-440: Is the usual logarithmic integral function ; the expression li( x ) in the second term should be considered as Ei( ρ log x ) , where Ei is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals. The final integral is equal to the series over the trivial zeros: Thus, Möbius inversion formula gives us valid for x > 1 , where
561-424: Is usually denoted as Π 0 ( x ) or J 0 ( x ) . It has jumps of 1 / n at prime powers p and it takes a value halfway between the two sides at the discontinuities of π ( x ) . That added detail is used because the function may then be defined by an inverse Mellin transform . Formally, we may define Π 0 ( x ) by where the variable p in each sum ranges over all primes within
594-461: The Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently). In 1899, de la Vallée Poussin proved that π ( x ) = li ( x ) + O ( x e −
627-445: The prime-counting function is the function counting the number of prime numbers less than or equal to some real number x . It is denoted by π ( x ) (unrelated to the number π ). A symmetric variant seen sometimes is π 0 ( x ) , which is equal to π ( x ) − 1 ⁄ 2 if x is exactly a prime number, and equal to π ( x ) otherwise. That is, the number of prime numbers less than x , plus half if x equals
660-414: The squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes , since they include two primes, or second numbers , by analogy with how "prime" means "first". The semiprimes less than 100 are: Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: The semiprimes are
693-534: The area of cryptography and number theory , most notably in public key cryptography , where they are used by RSA and pseudorandom number generators such as Blum Blum Shub . These methods rely on the fact that finding two large primes and multiplying them together (resulting in a semiprime) is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge , RSA Security offered prizes for
SECTION 20
#1732773365619726-424: The case k = 2 {\displaystyle k=2} of the k {\displaystyle k} - almost primes , numbers with exactly k {\displaystyle k} prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes). These are: A semiprime counting formula
759-663: The difference between π ( x ) and li( x ) : | π ( x ) − li ( x ) | ≤ 0.2593 x ( log x ) 3 / 4 exp ( − log x 6.315 ) for x ≥ 229. {\displaystyle {\bigl |}\pi (x)-\operatorname {li} (x){\bigr |}\leq 0.2593{\frac {x}{(\log x)^{3/4}}}\exp \left(-{\sqrt {\frac {\log x}{6.315}}}\right)\quad {\text{for }}x\geq 229.} For values of x that are not unreasonably large, li( x )
792-564: The factoring of specific large semiprimes and several prizes were awarded. The original RSA Factoring Challenge was issued in 1991, and was replaced in 2001 by the New RSA Factoring Challenge, which was later withdrawn in 2007. In 1974 the Arecibo message was sent with a radio signal aimed at a star cluster . It consisted of 1679 {\displaystyle 1679} binary digits intended to be interpreted as
825-431: The first used to prove the prime number theorem . They stem from the work of Riemann and von Mangoldt , and are generally known as explicit formulae . We have the following expression for the second Chebyshev function ψ : where Here ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero and one. The formula is valid for values of x greater than one, which
858-484: The number of natural numbers not greater than m which are divisible by none of the p i for any i ≤ n . Then Given a natural number m , if n = π ( √ m ) and if μ = π ( √ m ) − n , then Using this approach, Meissel computed π ( x ) , for x equal to 5 × 10 , 10 , 10 , and 10 . In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real m and for natural numbers n and k , P k ( m , n ) as
891-485: The number of numbers not greater than m with exactly k prime factors, all greater than p n . Furthermore, set P 0 ( m , n ) = 1 . Then where the sum actually has only finitely many nonzero terms. Let y denote an integer such that √ m ≤ y ≤ √ m , and set n = π ( y ) . Then P 1 ( m , n ) = π ( m ) − n and P k ( m , n ) = 0 when k ≥ 3 . Therefore, The computation of P 2 ( m , n ) can be obtained this way: where
924-630: The specified limits. We may also write where Λ is the von Mangoldt function and The Möbius inversion formula then gives where μ ( n ) is the Möbius function . Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function Λ , and using the Perron formula we have The Chebyshev function weights primes or prime powers p by log p : For x ≥ 2 , and Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting were
957-505: The sum is over prime numbers. On the other hand, the computation of Φ( m , n ) can be done using the following rules: Using his method and an IBM 701 , Lehmer was able to compute the correct value of π (10 ) and missed the correct value of π (10 ) by 1. Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat. Other prime-counting functions are also used because they are more convenient to work with. Riemann's prime-power counting function
990-489: The three functions π ( x ) , x / log x , and li( x ) compared at powers of 10. See also, and In the On-Line Encyclopedia of Integer Sequences , the π ( x ) column is sequence OEIS : A006880 , π ( x ) − x / log x is sequence OEIS : A057835 , and li( x ) − π ( x ) is sequence OEIS : A057752 . The value for π (10 )
1023-566: The value of Euler's totient function φ ( n ) {\displaystyle \varphi (n)} (the number of positive integers less than or equal to n {\displaystyle n} that are relatively prime to n {\displaystyle n} ) takes the simple form φ ( n ) = ( p − 1 ) ( q − 1 ) = n − ( p + q ) + 1. {\displaystyle \varphi (n)=(p-1)(q-1)=n-(p+q)+1.} This calculation
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1056-657: Was discovered by E. Noel and G. Panos in 2005. Let π 2 ( n ) {\displaystyle \pi _{2}(n)} denote the number of semiprimes less than or equal to n . Then π 2 ( n ) = ∑ k = 1 π ( n ) [ π ( n p k ) − k + 1 ] {\displaystyle \pi _{2}(n)=\sum _{k=1}^{\pi \left({\sqrt {n}}\right)}\left[\pi \left({\frac {n}{p_{k}}}\right)-k+1\right]} where π ( x ) {\displaystyle \pi (x)}
1089-539: Was originally computed by J. Buethe, J. Franke , A. Jost, and T. Kleinjung assuming the Riemann hypothesis . It was later verified unconditionally in a computation by D. J. Platt. The value for π (10 ) is by the same four authors. The value for π (10 ) was computed by D. B. Staple. All other prior entries in this table were also verified as part of that work. The values for 10 , 10 , and 10 were announced by David Baugh and Kim Walisch in 2015, 2020, and 2022, respectively. A simple way to find π ( x ) , if x
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