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Centered Nonagonal Number
A centered nonagonal number (or centered enneagonal number) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for ''n'' layers is given by the formula :Nc(n) = \frac. Multiplying the (''n'' - 1)th triangular number by 9 and then adding 1 yields the ''n''th centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number. Thus, the first few centered nonagonal numbers are : 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946. The list above includes the perfect numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime. Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, it follows that every even perfect nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Nonagonal Number
A centered nonagonal number (or centered enneagonal number) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for ''n'' layers is given by the formula :Nc(n) = \frac. Multiplying the (''n'' - 1)th triangular number by 9 and then adding 1 yields the ''n''th centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number. Thus, the first few centered nonagonal numbers are : 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946. The list above includes the perfect numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime. Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, it follows that every even perfect nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
190 (number)
190 (one hundred ndninety) is the natural number following 189 and preceding 191. In mathematics 190 is a triangular number, a hexagonal number, and a centered nonagonal number, the fourth figurate number (after 1, 28, and 91) with that combination of properties. It is also a truncated square pyramid number. Integers from 191 to 199 ;191 : 191 is a prime number. ;192 : 192 = 26 × 3 is a 3-smooth number, the smallest number with 14 divisors. ;193 : 193 is a prime number. ;194 : 194 = 2 × 97 is a Markov number, the smallest number written as the sum of three squares in five ways, and the number of irreducible representations of the Monster group. ;195 : 195 = 3 × 5 × 13 is the smallest number expressed as a sum of distinct squares in 16 different ways. ;196 :196 = 22 × 72 is a square number. ;197 :197 is a prime number and a Schröder–Hipparchus number. ;198 :198 = 2 × 32 × 11 is the smallest number written as the sum of four squares in ten ways :No integer factorial eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of The Theory Of Numbers
''History of the Theory of Numbers'' is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. The central topic of quadratic reciprocity and higher reciprocity laws is barely mentioned; this was apparently going to be the topic of a fourth volume that was never written . Volumes * Volume 1 - Divisibility and Primality - 486 pages * Volume 2 - Diophantine Analysis In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ... - 803 pages * Volume 3 - Quadratic and Higher Forms - 313 pages References * * * * * * * * * * * * External links History of the Theory of Numbers - Volume 1at the Internet Archive. History of the Theory of Numbers - Volu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sir Frederick Pollock, 1st Baronet
Sir Jonathan Frederick Pollock, 1st Baronet, PC (23 September 1783 – 28 August 1870) was a British lawyer and Tory politician. Background and education Pollock was the son of saddler David Pollock, of Charing Cross, London, and the elder brother of Field Marshal Sir George Pollock, 1st Baronet. An elder brother, Sir David Pollock, was a judge in India. The Pollock family were a branch of that family of Balgray, Dumfriesshire; David Pollock's father was a burgess of Berwick-upon-Tweed, and his grandfather a yeoman of Durham. His business as a saddler was given the official custom of the royal family. Sir John Pollock, 4th Baronet, great-great-grandson of David Pollock, stated in Time's Chariot (1950) that David was, 'perhaps without knowing it', Pollock of Balgray, the senior line of the family (Pollock of Pollock or Pollock of that ilk) having died out. Pollock was educated at St Paul's School and Trinity College, Cambridge. He was Senior Wrangler at Cambridge University. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28. This definition is ancient, appearing as early as Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p-1 for positive integer p—what is now called a Mersenne prime. Two millennia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
496 (number)
496 (four hundred ndninety-six) is the natural number following 495 and preceding 497. In mathematics 496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case. A triangular number and a hexagonal number, 496 is also a centered nonagonal number. Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular prime-indexed number is a prime number. It is the largest happy number less than 500. There is no solution to the equation φ(''x'') = 496, making 496 a nontotient. ''E''8 has real dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
136 (number)
136 (one hundred ndthirty six) is the natural number following 135 and preceding 137. In mathematics 136 is itself a factor of the Eddington number. With a total of 8 divisors, 8 among them, 136 is a refactorable number. It is a composite number. 136 is a centered triangular number and a centered nonagonal number. The sum of the ninth row of Lozanić's triangle is 136. 136 is a self-descriptive number in base 4, and a repdigit in base 16. In base 10, the sum of the cubes of its digits is 1^3 + 3^3 + 6^3 = 244. The sum of the cubes of the digits of 244 is 2^3 + 4^3 + 4^3 = 136. 136 is a triangular number, because it's the sum of the first 16 positive integers. In the military * Force 136 branch of the British organization, the Special Operations Executive (SOE), in the South-East Asian Theatre of World War II * USNS ''Mission Soledad'' (T-AO-136) was a United States Navy ''Mission Buenaventura''-class fleet oiler during World War II * USS ''Admirable'' (AM-136) was a United ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered ''k''-gonal number contains ''k'' more dots than the previous layer. Examples Each centered ''k''-gonal number in the series is ''k'' times the previous triangular number, plus 1. This can be formalized by the expression \frac +1, where ''n'' is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression \frac +1. These series consist of the *centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (), *centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
91 (number)
91 (ninety-one) is the natural number following 90 and preceding 92. In mathematics 91 is: * the twenty-seventh distinct semiprime and the second of the form (7×q). * a triangular number. * a hexagonal number, one of the few such numbers to also be a centered hexagonal number. * a centered nonagonal number. * a centered cube number. * a square pyramidal number, being the sum of the squares of the first six integers. * the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): . (See 1729 for more details). This implies that 91 is the second cabtaxi number. * the smallest positive integer expressible as a sum of six distinct squares: . * The only other ways to write 91 as a sum of distinct squares are: and * . * the smallest pseudoprime satisfying the congruence .Friedman, ErichWhat's Special About This Number? * a repdigit in base 9 (1119). * palindr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
