|
71 (number)
71 (seventy-one) is the natural number following 70 and preceding 72. __TOC__ In mathematics 71 is the 20th prime number. Because both rearrangements of its digits (17 and 71) are prime numbers, 71 is an emirp and more generally a permutable prime. 71 is a centered heptagonal number. It is a regular prime, a Ramanujan prime, a Higgs prime, and a good prime. It is a Pillai prime, since 9!+1 is divisible by 71, but 71 is not one more than a multiple of 9. It is part of the last known pair (71, 7) of Brown numbers, since 71^=7!+1. 71 is the smallest of thirty-one discriminants of imaginary quadratic fields with class number of 7, negated (see also, Heegner number In number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from int ...s). 71 is the largest number which occurs as a prim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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, or , 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 the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, 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 primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higgs Prime
A Higgs prime, named after Denis Higgs, is a prime number with a totient (one less than the prime) that evenly divides the square of the product of the smaller Higgs primes. (This can be generalized to cubes, fourth powers, etc.) To put it algebraically, given an exponent ''a'', a Higgs prime ''Hp''''n'' satisfies : \phi(Hp_n), \prod_^ ^a\mboxHp_n > Hp_ where Φ(''x'') is Euler's totient function. For squares, the first few Higgs primes are 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, ... . So, for example, 13 is a Higgs prime because the square of the product of the smaller Higgs primes is 5336100, and divided by 12 this is 444675. But 17 is not a Higgs prime because the square of the product of the smaller primes is 901800900, which leaves a remainder of 4 when divided by 16. From observation of the first few Higgs primes for squares through seventh powers, it would seem more compact to list those primes that are not Higgs primes: Observation further revea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersingular Prime (moonshine Theory)
In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group M, which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31; as well as 41, 47, 59, and 71 . The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73. Supersingular primes are related to the notion of supersingular elliptic curves as follows. For a prime number p, the following are equivalent: # The modular curve X_0^+(p) = X_0(p)/w_p, where w_p is the Fricke involution of X_0(p), has genus zero. # Every supersingular elliptic curve in characteristic p can be defined over the prime subfield \mathbb_p. # The order of the Monster group is divisible by p. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying the first condition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sporadic Simple Group
In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The mentioned classification theorem states that the list of finite simple groups consists of 18 countably infinite families plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. The Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type, in which case there would be 27 sporadic groups. The monster group, or ''friendly giant'', is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Names Five of the sporadic groups were discovered by Émile Mathieu in the 1860s and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heegner Number
In number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ..., a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, the ring of algebraic integers of \Q\left sqrt\right/math> has unique factorization. The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the (Baker–) Stark–Heegner theorem there are precisely nine Heegner numbers: This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated that the gap in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a Field extension, subfield of the field of the real numbers. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important. Ring of integers Discriminant For a nonzero square free integer , the Discriminant of an algebraic number field, discriminant of the quadratic field is |
Brown Number
Brocard's problem is a problem in mathematics that seeks integer values of n such that n!+1 is a perfect square, where n! is the factorial. Only three values of n are known — 4, 5, 7 — and it is not known whether there are any more. More formally, it seeks pairs of integers n and m such thatn!+1 = m^2.The problem was posed by Henri Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Srinivasa Ramanujan. Brown numbers Pairs of the numbers (n,m) that solve Brocard's problem were named Brown numbers by Clifford A. Pickover in his 1995 book ''Keys to Infinity'', after learning of the problem from Kevin S. Brown. As of October 2022, there are only three known pairs of Brown numbers: based on the equalities Paul Erdős conjectured that no other solutions exist. Computational searches up to one quadrillion have found no further solutions. Connection to the abc conjecture It would follow from the abc conjecture that there are only finitely many B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pillai Prime
In number theory, a Pillai prime is a prime number ''p'' for which there is an integer ''n'' > 0 such that the factorial of ''n'' is one less than a multiple of the prime, but the prime is not one more than a multiple of ''n''. To put it algebraically, n! \equiv -1 \mod p but p \not\equiv 1 \mod n. The first few Pillai primes are : 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers. Their infinitude has been proven several times, by Subbarao, Erdős, and Hardy & Subbarao. References *. *. *https://planetmath.org/pillaiprime, PlanetMath PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org ... Classes of prime numbers Eponymous numbers in mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good Prime
A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality :p_n^2 > p_ \cdot p_ for all 1 ≤ ''i'' ≤ ''n''−1, where ''pk'' is the ''k''th prime. Example: the first primes are 2, 3, 5, 7 and 11. Since for 5 both the conditions :5^2 >3 \cdot 7 :5^2 >2 \cdot 11 are fulfilled, 5 is a good prime. There are infinitely many good primes. The first good primes are: : 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853, 929, 937, 967 . An alternative version takes only ''i'' = 1 in the definition. With that there are more good primes: : 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179, 191, 197 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Prime
In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function. Origins and definition In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is: : \pi(x) - \pi\left( \frac x 2 \right) \ge 1,2,3,4,5,\ldots \text x \ge 2, 11, 17, 29, 41, \ldots \text where \pi(x) is the prime-counting function, equal to the number of primes less than or equal to ''x''. The converse of this result is the definition of Ramanujan primes: :The ''n''th Ramanujan prime is the least integer ''Rn'' for which \pi(x) - \pi(x/2) \ge n, for all ''x'' ≥ ''Rn''. In other words: Ramanujan primes are the least integers ''Rn'' for which there are at least ''n'' primes between ''x'' and ''x''/2 for all ''x'' ≥ ''Rn''. The first five Ramanujan pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers. The first few regular odd primes are: : 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... . History and motivation In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent ''p'' if ''p'' is regular. This focused attention on the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent ''p'', if is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either or fails to be an irregular pair. ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
