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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Goldbach's Conjecture, Twin Prime Conjecture, Goldbach's Weak Conjecture, Schinzel's Hypothesis H, Landau's Problems, Mersenne Conjectures, Artin's Conjecture on Primitive Roots, Pólya Conjecture, Polignac's Conjecture, Cramér's Conjecture, Gilbreath's Conjecture, Chinese Hypothesis, Andrica's Conjecture, Elliott-halberstam Conjecture, Second Hardy-littlewood Conjecture, Bateman-Horn Conjecture, Bunyakovsky Conjecture, Agoh-giuga Conjecture, Lemoine's Conjecture, Legendre's Conjecture, Legendre's Constant, Redmond-sun Conjecture, Brocard's Conjecture, Waring's Prime Number Conjecture, Grimm's Conjecture, First Hardy-Littlewood Conjecture. Excerpt:In number theory the Agoh Giuga conjecture on the Bernoulli numbers B k postulates that p is a prime number if and only if The conjecture as stated is due to Takashi Agoh (1990); an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that p is prime iff which may also be written as It is trivial to show that p being prime is sufficient for the second equivalence to hold, since if p is prime, Fermat's little theorem states that for , and the equivalence follows, since The statement is still a conjecture since it has not yet been proven that if a number n is not prime (that is, n is composite ), then the formula does not hold. It has been shown that a composite number n satisfies the formula iff it is both a Carmichael number and a Giuga number , and that if such a number exists, it has at least 13,800 digits (Borwein, Borwein, Borwein, Girgensohn 1996). The Agoh-Giuga conjecture bears a similarity to Wilson's theorem , which has been proven to be true. Wilson's theorem states that a number p is prime iff which may also be written as or, for odd prime p and, for even prime p=2 So, the truth of the Agoh-Giuga conjec... ... Read more

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High Quality Content by WIKIPEDIA articles! A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn, is the difference between the (n + 1)-th and the n-th prime number, i.e. gn = pn + 1 ? pn.In reality, prime gaps of P numbers can occur at numbers much smaller than P#. For instance, the smallest sequence of 71 consecutive composite numbers occurs between 31398 and 31468, whereas 71# has twenty-seven digits - its full decimal expansion being 557940830126698960967415390. Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the maximum prime gap to the integers involved also increases as larger and larger numbers and gaps are encountered. ... Read more

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Prime Number Theorem. Prime Number, Abstract Analytic Number Theory, Landau Prime Ideal Theorem, Prime Gap, Twin Prime Conjecture, Number Theory, Multiplicative Number Theory, On the Number of Primes Less Than a Given Magnitude, Explicit Formula, Arithmetic Progression, Article Sources and Contributors, Image Sources, Licenses and Contributors ... Read more

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High Quality Content by WIKIPEDIA articles! A twin prime is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), and (skipping quite a few), (821, 823). Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin. ... Read more