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Prime formulas

There exist a variety of formulas for either producing the th prime as a function of or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula (Dudley 1969; Ribenboim 1996, p. 186). There also exist simple prime-generating polynomials that generate only primes for the first (possibly large) number of integer values.There are also many beautiful formulas involving prime sums and prime products that can be done in closed form.Considering examples of formulas that produce only prime numbers (although not necessarily the complete set of prime numbers ), there exists a constant (OEIS A051021) known as Mills' constant such that(1)where is the floor function, is prime for all (Ribenboim 1996, p. 186). The first few values of are 2, 11, 1361, 2521008887, ... (OEIS A051254). It..

Sum of prime factors

Let be the sum of prime factors (with repetition) of a number . For example, , so . Then for , 2, ... is given by 0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, ... (OEIS A001414). The sum of prime factors function is also known as the integer logarithm.The high-water marks are 0, 2, 3, 4, 5, 7, 11, 13, 17, ..., which occur at positions 1, 2, 3, 4, 5, 7, 11, 13, 17, ... (OEIS A046022), which, with the exception of the first term, correspond exactly to the actual values of the high-water marks.If is considered to be 0 for a prime, then the sequence of high-water marks is 0, 4, 5, 6, 7, 9, 10, 13, 15, 19, 21, 25, 31, 33, ... (OEIS A088685), which occur at positions 1, 4, 6, 8, 10, 14, 21, 22, 26, 34, 38, 46, 58, ... (OEIS A088686). Rather amazingly, if the first 7 terms are dropped, then the last digit of the high-water marks and the last digit of their positions fall into one of the four patterns , (3, 2), (5, 6), or (9, 4) (A. Jones, pers. comm., October 5, 2003).Now consider iterating..

Prime representation

Let , , and denote positive integers satisfying(i.e., both pairs are relatively prime), and suppose every prime with is expressible if the form for some integers and . Then every prime such that and is expressible in the form for some integers and (Halter-Koch 1993, Williams 1991).prime formrepresentation

Prime sums

Let(1)be the sum of the first primes (i.e., the sum analog of the primorial function). The first few terms are 2, 5, 10, 17, 28, 41, 58, 77, ... (OEIS A007504). Bach and Shallit (1996) show that(2)and provide a general technique for estimating such sums.The first few values of such that is prime are 1, 2, 4, 6, 12, 14, 60, 64, 96, 100, ... (OEIS A013916). The corresponding values of are 2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, ... (OEIS A013918).The first few values of such that are 1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, ... (OEIS A045345). The corresponding values of are 2, 874, 5830, 2615298, 712377380, 86810649294, 794712005370, 105784534314378, 92542301212047102, ... (OEIS A050247; Rivera), and the values of are 2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, ... (OEIS A050248; Rivera).In 1737, Euler showed that the harmonic seriesof primes, (i.e., sum of the reciprocals of the primes) diverges(3)(Nagell..

Mertens theorem

Consider the Euler product(1)where is the Riemann zeta function and is the th prime. , but taking the finite product up to , premultiplying by a factor , and letting gives(2)(3)where is the Euler-Mascheroni constant (Havil 2003, p. 173). This amazing result is known as the Mertens theorem.At least for , the sequence of finite products approaches strictly from above (Rosser and Schoenfeld 1962). However, it is highly likely that the finite product is less than its limiting value for infinitely many values of , which is usually the case for any such inequality due to the presence of zeros of on the critical line . An example is Littlewood's famous proof that the sense of the inequality , where is the prime counting function and is the logarithmic integral, reverses infinitely often. While Rosser and Schoenfeld (1962) suggest that "perhaps one can extend [this] result to show that [the Mertens inequality] fails for large ; we have not investigated..

Chebyshev functions

The two functions and defined below are known as the Chebyshev functions.The function is defined by(1)(2)(3)(Hardy and Wright 1979, p. 340), where is the th prime, is the prime counting function, and is the primorial. This function has the limit(4)and the asymptotic behavior(5)(Bach and Shallit 1996; Hardy 1999, p. 28; Havil 2003, p. 184). The notation is also commonly used for this function (Hardy 1999, p. 27).The related function is defined by(6)(7)where is the Mangoldt function (Hardy and Wright 1979, p. 340; Edwards 2001, p. 51). Here, the sum runs over all primes and positive integers such that , and therefore potentially includes some primes multiple times. A simple and beautiful formula for is given by(8)i.e., the logarithm of the least common multiple of the numbers from 1 to (correcting Havil 2003, p. 184). The values of for , 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, ... (OEIS A003418;..

Mertens' second theorem

Mertens' second theorem states that the asymptotic form of the harmonic series for the sum of reciprocal primes is given bywhere is a prime, is a constant known as the Mertens constant, and is a Landau symbol.

Mertens constant

The Mertens constant , also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant (Bach and Shallit 1996, p. 234), or Kronecker's constant (Schroeder 1997), is a constant related to the twin primes constant and that appears in Mertens' second theorem,(1)where the sum is over primes and is a Landau symbol. This sum is the analog of(2)where is the Euler-Mascheroni constant (Gourdon and Sebah).The constant is given by the infinite sum(3)where is the Euler-Mascheroni constant and is the th prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979; Le Lionnais 1983; Ellison and Ellison 1985), or by the limit(4)According to Lindqvist and Peetre (1997), this was shown independently by Meisselin 1866 and Mertens (1874). Formula (3) is equivalent to(5)which follows from (4) using the Mercator series for with . is also given by the rapidly converging series(6)where is the Riemann zeta function, and is the Möbius..

Brun's constant

The number obtained by adding the reciprocals of the odd twinprimes,(1)By Brun's theorem, the series converges to a definite number, which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1989, p. 201). By contrast, the series of all prime reciprocals diverges to infinity, as follows from the Mertens second theorem by letting (which provides a stronger characterization of the divergence than Euler's proof that , obtained more than a century before Mertens' proof).Shanks and Wrench (1974) used all the twin primes among the first 2 million numbers. Brent (1976) calculated all twin primes up to 100 billion and obtained (Ribenboim 1989, p. 146)(2)assuming the truth of the first Hardy-Littlewood conjecture. Using twin primes up to , Nicely (1996) obtained(3)(Cipra 1995, 1996), in the process discovering a bug in Intel's® PentiumTM microprocessor. Using twin primes up to , Nicely..

Harmonic series of primes

Like the entire harmonic series, the harmonicseries(1)taken over all primes also diverges, as first shown by Euler in 1737 (Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17 and 22; Wells 1986, p. 41; Havil 2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (OEIS A046024) primes.Its asymptotic behavior is given by(2)where(3)(OEIS A077761) is the Mertens constant (Hardy and Wright 1979, p. 351; Hardy 1999, p. 50; Havil 2003, p. 64).

Prime zeta function

The prime zeta function(1)where the sum is taken over primes is a generalizationof the Riemann zeta function(2)where the sum is over all positive integers. In other words, the prime zeta function is the Dirichlet generating function of the characteristic function of the primes . is illustrated above on positive the real axis, where the imaginary part is indicated in yellow and the real part in red. (The sign difference in the imaginary part compared to the plot appearing in Fröberg is presumably a result of the use of a different convention for .)Various terms and notations are used for this function. The term "prime zeta function" and notation were used by Fröberg (1968), whereas Cohen (2000) uses the notation .The series converges absolutely for , where , can be analytically continued to the strip (Fröberg 1968), but not beyond the line (Landau and Walfisz 1920, Fröberg 1968) due to the clustering of singular..

Primorial prime

Primorial primes are primes of the form , where is the primorial of . A coordinated search for such primes is being conducted on PrimeGrid. is prime for , 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... (OEIS A057704; Guy 1994, pp. 7-8; Caldwell 1995). These correspond to with , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, ... (OEIS A006794). The largest known primorial primes as of Nov. 2015 are summarized in the following table (Caldwell).digitsdiscoverer6845Dec. 1992365851PrimeGrid (Dec. 20, 2010)476311PrimeGrid (Mar. 5, 2012) (also known as a Euclid number) is prime for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... (OEIS A014545; Guy 1994, Caldwell 1995, Mudge 1997). These correspond to with , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547,..

Primorial

Let be the th prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by(1)The values of for , 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (OEIS A002110).It is sometimes convenient to define the primorial for values other than just the primes, in which case it is taken to be given by the product of all primes less than or equal to , i.e.,(2)where is the prime counting function. For , 2, ..., the first few values of are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS A034386).The logarithm of is closely related to the Chebyshev function , and a trivial rearrangement of the limit(3)gives(4)(Ruiz 1997; Finch 2003, p. 14; Pruitt), where eis the usual base of the natural logarithm.

Prime products

The product of primes(1)with the th prime, is called the primorial function, by analogy with the factorial function. Its logarithm is closely related to the Chebyshev function .The zeta-regularized product over allprimes is given by(2)(3)(Muñoz Garcia and Pérez-Marco 2003, 2008), answering the question posed by Soulé et al. (1992, p. 101). A derivation proceeds by algebraic manipulation of the prime zeta function and gives the more general results(4)and(5)(Muñoz Garcia and Pérez-Marco 2003).Mertens theorem states that(6)where is the Euler-Mascheroni constant, and a closely related result is given by(7)There are amazing infinite product formulas forprimes given by(8)(Ramanujan 1913-1914; Le Lionnais 1983, p. 46) and(9)(OEIS A082020; Ramanujan 1913-1914).More general formulas are given by(10)where is the Riemann zeta function and by the Euler product(11)Named prime..

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