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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..

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..

Fermat's 4n+1 theorem

Fermat's theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number can be represented in an essentially unique manner (up to the order of addends) in the form for integer and iff or (which is a degenerate case with ). The theorem was stated by Fermat, but the first published proof was by Euler.The first few primes which are 1 or 2 (mod 4) are 2, 5, 13, 17, 29, 37, 41, 53, 61, ... (OEIS A002313) (with the only prime congruent to 2 mod 4 being 2). The numbers such that equal these primes are (1, 1), (1, 2), (2, 3), (1, 4), (2, 5), (1, 6), ... (OEIS A002331 and A002330).The theorem can be restated by lettingthen all relatively prime solutions to the problem of representing for any integer are achieved by means of successive applications of the genus theorem and composition theorem...

Sierpiński's prime sequence theorem

For any , there exists a such that the sequencewhere , 2, ... contains at least primes.

Euler's 6n+1 theorem

Euler's theorem states that every prime of the form , (i.e., 7, 13, 19, 31, 37, 43, 61, 67, ..., which are also the primes of the form ; OEIS A002476) can be written in the form with and positive integers.The first few positive integers that can be represented in this form (with ) are 4, 7, 12, 13, 16, 19, ... (OEIS A092572), summarized in the following table together with their representations.4(1, 1)7(2, 1)12(3, 1)13(1, 2)16(2, 2)19(4, 1)21(3, 2)28(1, 3), (4, 2), (5, 1)31(2, 3)Restricting solutions such that (i.e., and are relatively prime), the numbers that can be represented as are 4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, ... (OEIS A092574), as summarized in the following table. with 4(1, 1)7(2, 1)12(3, 1)13(1, 2)19(4, 1)21(3, 2)28(1, 3), (5, 1)31(2, 3)37(5, 2)

Schnirelmann's theorem

There exists a positive integer such that every sufficiently large integer is the sum of at most primes. It follows that there exists a positive integer such that every integer is a sum of at most primes. The smallest proven value of is known as the Schnirelmann constant.Schnirelmann's theorem can be proved using Mann's theorem,although Schnirelmann used the weaker inequalitywhere , , and is the Schnirelmann density. Let be the set of primes, together with 0 and 1, and let . Using a sophisticated version of the inclusion-exclusion principle, Schnirelmann showed that although , . By repeated applications of Mann's theorem, the sum of copies of satisfies . Thus, if , the sum of copies of has Schnirelmann density 1, and so contains all positive integers.

Choquet theory

Erdős proved that there exist at least one prime of the form and at least one prime of the form between and for all .

Chen's theorem

Every "large" even number may be written as where is a prime and is the set of primes and semiprimes .

Levy's conjecture

Levy (1963) noted that(1)(2)and from this observation, conjectured that all odd numbers are the sum of a prime plus twice a prime. This conjecture is a stronger version of the weak Goldbach conjecture and has been verified up to (Corbit 1999).The number of ways to express as for and primes and , 2, ... are 0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, ... (OEIS A046927).

Sierpiński's composite number theorem

As proved by Sierpiński (1960), there exist infinitely many positive odd numbers such that is composite for every . Numbers with this property are called Sierpiński numbers of the second kind, and analogous numbers with the plus sign replaced by a minus are called Riesel numbers. It is conjectured that the smallest value of for a Sierpiński number of the second kind is (although a handful of smaller candidates remain to be eliminated) and that the smallest Riesel number is .

Selfridge's conjecture

There exist infinitely many with for all , where is the th prime. Also, there exist infinitely many such that for all .

Prime partition

A prime partition of a positive integer is a set of primes which sum to . For example, there are three prime partitions of 7 sinceThe number of prime partitions of , 3, ... are 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, ... (OEIS A000607). If for prime and for composite, then the Euler transform gives the number of partitions of into prime parts (Sloane and Plouffe 1995, p. 21).The minimum number of primes needed to sum to , 3, ... are 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, ... (OEIS A051034). The maximum number of primes needed to sum to is just , 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... (OEIS A004526), corresponding to a representation in terms of all 2s for an even number or one 3 and the rest 2s for an odd number.The numbers which can be represented by a single prime are obviously the primes themselves. Composite numbers which can be represented as the sum of two primes are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ... (OEIS A051035), and composite..

Mills' theorem

Mills (1947) proved the existence of a real constant such that(1)is prime for all integers , where is the floor function. Mills (1947) did not, however, determine , or even a range for .A generalization of Mills' theorem to an arbitrary sequence of positiveintegers is given as an exercise by Ellison and Ellison (1985).The least such that is prime for all integers is known as Mills' constant.Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let be the th prime, then there exists a constant such that(2)for all . This has more recently been strengthened to(3)(Mozzochi 1986). If the Riemann hypothesisis true, then Cramér (1937) showed that(4)(Finch 2003).Hardy and Wright (1979) and Ribenboim (1996) point out that, despite the beauty of such prime formulas, they do not have any practical consequences. In fact, unless the exact value of is known, the primes themselves must be known in advance to determine..

Mills' constant

Mills' theorem states that there exists a real constant such that is prime for all positive integers (Mills 1947). While for each value of , there are uncountably many possible values of such that is prime for all positive integers (Caldwell and Cheng 2005), it is possible to define Mills' constant as the least such thatis prime for all positive integers , giving a value of(OEIS A051021). is therefore given by the next prime after , and the values of are known as Mills' primes (Caldwell and Cheng 2005).Caldwell and Cheng (2005) computed more than 6850 digits of assuming the truth of the Riemann hypothesis. Proof of primality of the 13 Mills prime in Jul. 2013 means that approximately digits are now known.It is not known if is irrational.

Landau's problems

Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely: 1. The Goldbach conjecture, 2. Twin prime conjecture, 3. Legendre's conjecture that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and 4. The conjecture that there are infinitely many primes of the form (Euler 1760; Mirsky 1949; Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208). The first few such primes are 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496). Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest primes between and for , 2, ..., are 2, 5, 11,..

Goldbach conjecture

Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes" (Goldbach 1742; Dickson 2005, p. 421). Note that Goldbach considered the number 1 to be a prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes. Two primes such that for a positive integer are sometimes called a Goldbach partition (Oliveira e Silva).According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and..

Idoneal number

An idoneal number, also called a suitable number or convenient number, is a positive integer for which the fact that a number is a monomorph (i.e., is expressible in only one way as where is relatively prime to ) guarantees it to be a prime, prime power, or twice one of these. The numbers are also called Euler's idoneal numbers or suitable numbers.A positive integer is idoneal iff it cannot be written as for integer , , and with .The 65 idoneal numbers found by Gauss and Euler and conjectured to be the only such numbers (Shanks 1969) are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 (OEIS A000926). It is known that if any other idoneal numbers exist, there can be only one more...

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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