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Cuban Prime
A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers ''x'' and ''y''. First series This is the first of these equations: :p = \frac,\ x = y + 1,\ y>0, i.e. the difference between two successive cubes. The first few cuban primes from this equation are : 7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227 The formula for a general cuban prime of this kind can be simplified to 3y^2 + 3y + 1. This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal. the largest known has 65537 digits with y = 100000845^, found by Jens Kruse Andersen. Second series The second of these equations is: :p = \frac,\ x = y + 2,\ y>0. ... [...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 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 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 method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7 (number)
7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. As an early prime number in the series of positive integers, the number seven has greatly symbolic associations in religion, mythology, superstition and philosophy. The seven Classical planets resulted in seven being the number of days in a week. It is often considered lucky in Western culture and is often seen as highly symbolic. Unlike Western culture, in Vietnamese culture, the number seven is sometimes considered unlucky. It is the first natural number whose pronunciation contains more than one syllable. Evolution of the Arabic digit In the beginning, Indians wrote 7 more or less in one stroke as a curve that looks like an uppercase vertically inverted. The western Ghubar Arabs' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arabs developed the digit fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19 (number)
19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number. Mathematics 19 is the eighth prime number, and forms a sexy prime with 13, a twin prime with 17, and a cousin prime with 23. It is the third full reptend prime, the fifth central trinomial coefficient, and the seventh Mersenne prime exponent. It is also the second Keith number, and more specifically the first Keith prime. * 19 is the maximum number of fourth powers needed to sum up to any natural number, and in the context of Waring's problem, 19 is the fourth value of g(k). * The sum of the squares of the first 19 primes is divisible by 19. *19 is the sixth Heegner number. 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of nine total. * 19 is the third centered triangular number as well as the third centered hexagonal number. : The 19th triangular number is 190, equivalently the sum of the first 19 non-zero integers, that is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
37 (number)
37 (thirty-seven) is the natural number following 36 and preceding 38. In mathematics 37 is the 12th prime number and the third unique prime in decimal. 37 is the first irregular prime, and the third isolated prime without a twin prime. It is also the third cuban prime, the fourth emirp, and the fifth lucky prime. *37 is the third star number and the fourth centered hexagonal number. *The sum of the squares of the first 37 primes is divisible by 37. *Every positive integer is the sum of at most 37 fifth powers (see Waring's problem). *37 appears in the Padovan sequence, preceded by the terms 16, 21, and 28 (it is the sum of the first two of these). *Since the greatest prime factor of 372 + 1 = 1370 is 137, which is substantially more than 37 twice, 37 is a Størmer number. In base-ten, 37 is a permutable prime with 73, which is the 21st prime number. By extension, the mirroring of their digits and prime indexes makes 73 the only Sheldon prime. In moonshine theory, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
61 (number)
61 (sixty-one) is the natural number following 60 and preceding 62. In mathematics 61 is: *the 18th prime number. *a twin prime with 59. *a cuban prime of the form ''p'' = , where ''x'' = ''y'' + 1. *the smallest ''proper prime'', a prime ''p'' which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repeating sequence with length ''p'' − 1. In such primes, each digit 0, 1, ..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, times). *the exponent of the 9th Mersenne prime. (261 − 1 = ) *the sum of two squares, 52 + 62. *a centered square number. *a centered hexagonal number. *a centered decagonal number. *the sixth Euler zigzag number (or Up/down number). *a unique prime in base 14, since no other prime has a 6-digit period in base 14. *a Pillai prime since 8! + 1 is divisible by 61 but 61 is not one more than a multiple of 8. *a Keith number, because it recurs in a Fibonacci-like sequence started from i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
127 (number)
127 (one hundred ndtwenty-seven) is the natural number following 126 and preceding 128. It is also a prime number. In mathematics *As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also the largest known mersenne prime exponent for a Mersenne number, 2^-1, which is also a Mersenne prime. It was discovered by Édouard Lucas in 1876 and held the record for the largest known prime for 75 years. **2^-1 is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime. ** Furthermore, 127 is equal to 2^-1, and 7 is equal to 2^-1, and 3 is the smallest Mersenne prime, making 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime. *There are a total of 127 prime numbers between 2,000 and 3,000. *127 is also a cuban prime of the form p=\frac, x=y+1. The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime. 127 is greate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Hexagonal Number
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers: : Centered hexagonal numbers should not be confused with cornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex. The sequence of hexagonal numbers starts out as follows : : 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919. Formula The th centered hexagonal number is given by the formula :H(n) = n^3 - (n-1)^3 = 3n(n-1)+1 = 3n^2 - 3n +1. \, Expressing the formula as :H(n) = 1+6\left(\frac\right) shows that the centered hexagonal number for is 1 more than 6 times the th triangular number. In the opposite direction, the ''index'' corresponding to the centered he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube (arithmetic)
In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or . The cube is also the number multiplied by its square: :. The ''cube function'' is the function (often denoted ) that maps a number to its cube. It is an odd function, as :. The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. The graph of the cube function is known as the cubic parabola. Because the cube function is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry. In integers A cube number, or a perfect cube, or sometimes just a cube, is a number which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree three, and a real function. In particular, the domain and the codomain are the set of the real numbers. Setting produces a cubic equation of the form :ax^3+bx^2+cx+d=0, whose solutions are called roots of the function. A cubic function has either one or three real roots (which may not be distinct); all odd-degree polynomials have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Prime Numbers
This is a list of articles about prime numbers. A prime number (or ''prime'') is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. 1 is neither prime nor composite. The first 1000 prime numbers The following table lists the first 1000 primes, with 20 columns of consecutive primes in each of the 50 rows. . The Goldbach conjecture verification project reports that it has computed all primes below 4×10. That means 95,676,260,903,887,607 primes (nearly 10), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes below a given value) faster than computing the primes. This has been used to c ... [...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 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 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 method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Pages
The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms. , the 5,000th prime has around 412,000 digits.. Retrieved on 2018-02-12. The PrimePages has articles on primes and primality testing. It includes "The Prime Glossary" with articles on hundreds of glosses related to primes, and "Prime Curios!" with thousands of curios about specific numbers. The database started as a list of titanic primes (primes with at least 1000 decimal digits) by Samuel Yates. In subsequent years, the whole top-5,000 has consisted of gigantic primes (primes with at least 10,000 decimal digits). Primes of special forms are kept on the current lists if they are titanic and in the top-20 or top-5 for their form. See also *List of prime numbers This is a list of articles about pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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