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Centered Square Number
In elementary number theory, a centered square number is a Centered polygonal number, centered figurate number that gives the number of dots in a Square (geometry), square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given Taxicab geometry, city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties. The figures for the first four centered square numbers are shown below: : Each centered square number is the sum of successive squares. Example: as shown in the following figure of Floyd's triangle, 25 is a centered square number, and is the sum of the square 16 (yellow rhombus formed by shearing a square) and of the next smaller sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floyd's Triangle
Floyd's triangle is a triangular array of natural numbers used in computer science education. It is named after Robert W. Floyd, Robert Floyd. It is defined by filling the rows of the triangle with consecutive numbers, starting with a 1 in the top left corner: The problem of writing a computer program to produce this triangle has been frequently used as an exercise or example for beginning computer programmers, covering the concepts of text formatting and simple Control flow#Loops, loop constructs. Properties *The numbers along the left edge of the triangle are the lazy caterer's sequence and the numbers along the right edge are the triangular numbers. The ''n''th row sums to , the constant of an magic square . *Summing up the row sums in Floyd's triangle reveals the doubly triangular numbers, triangular numbers with an index that is triangular.. 1 = 1 = T(T(1)) 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary 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, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry). Questions in number theory can often be understood through the study of analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is that it deals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Triangular Number
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers. This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to n. The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue). Properties *The gnomon of the ''n''-th centered triangular number, corresponding to the (''n'' + 1)-th triangular layer, is: ::C_ - C_ = 3(n+1). *The ''n''-th centered triangular number, corresponding to ''n'' layers ''plus'' the center, is given by the formula: ::C_ = 1 + 3 \frac = \frac. *Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
365 (number)
365 (three hundred ndsixty-five) is the natural number following 364 and preceding 366. Mathematics 365 is a semiprime centered square number. It is also the fifth 38 -gonal number. For multiplication, it is calculated as 5 \times 73. Both 5 and 73 are prime numbers. It is the smallest number that has more than one expression as a sum of consecutive square numbers: :365 = 13^2 + 14^2 :365 = 10^2 + 11^2 + 12^2 There are no known primes with period 365, while at least one prime with each of the periods 1 to 364 is known. Timekeeping There are 365.2422 solar days in the mean tropical year. Several solar calendars have a year containing 365 days. Related to this, in Ontario, the driver's license A driver's license, driving licence, or driving permit is a legal authorization, or the official document confirming such an authorization, for a specific individual to operate one or more types of motorized vehicles—such as motorcycles, ca ... learner's permit used to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
313 (number)
313 (three hundred ndthirteen) is the natural number following 312 and preceding 314. In mathematics 313 is: *a twin prime with 311 *a centered square number *a full reptend prime (and the smallest number which is a full reptend prime in base 10 but not in base 2 to 9) *a Pythagorean prime *a regular prime *a palindromic prime in both decimal and binary. *a truncatable prime *a weakly prime in base 5 *a happy number *an Armstrong number - in base 4 ( 3×42 + 1×41 + 3×40 = 33 + 13 + 33 ) *an index of a prime Lucas number. *a palindromic number in base 2 (100111001) In religion * The number of times the word pray appears in the King James Bible. * The number of soldiers that Muhammad had with him in the first battle fought by the Muslims, the Battle of Badr The Battle of Badr or sometimes called The Raid of Badr ( ; ''Ghazwahu Badr''), also referred to as The Day of the Criterion (, ; ''Yawm al-Furqan'') in the Qur'an and by Muslims, was fought on 13 March 624 CE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
221 (number)
221 (two hundred ndtwenty-one) is the natural number following 220 and preceding 222. In mathematics Its factorization as 13 × 17 makes 221 the product of two consecutive prime numbers, the sixth smallest such product. 221 is a centered square number, an Ulam number, and a brilliant number, meaning that its prime factors have the same amount of digits. In Other Fields The year 221 BC marked the end of the Warring States period and the beginning of the Qin dynasty in China. 221b Baker Street is the address of the fictional detective Sherlock Holmes Sherlock Holmes () is a Detective fiction, fictional detective created by British author Arthur Conan Doyle. Referring to himself as a "Private investigator, consulting detective" in his stories, Holmes is known for his proficiency with obser ..., created by Sir Arthur Conan Doyle. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
181 (number)
181 (one hundred ndeighty-one) is the natural number following 180 and preceding 182. In mathematics 181 is prime, and a palindromic, strobogrammatic, and dihedral number in decimal. 181 is a Chen prime. 181 is a twin prime with 179, equal to the sum of ''five'' consecutive prime numbers: 29 + 31 + 37 + 41 + 43. 181 is the difference of two consecutive square numbers 912 – 902, as well as the sum of two consecutive squares: 92 + 102. As a '' centered polygonal number'', 181 is: 181 is also a centered ( hexagram) '' star number'', as in the game of Chinese checkers. Specifically, 181 is the 42nd prime number and ''16th'' ''full reptend prime'' in decimal, where multiples of its reciprocal \tfrac inside a prime reciprocal magic square repeat 180 digits with a magic sum M of 810; this value is one less than 811, the 141st prime number and ''49th'' full reptend prime (or equivalently ''long prime'') in decimal whose reciprocal repeats 810 digits. Whil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
145 (number)
145 (one hundred ndforty-five) is the natural number following 144 and preceding 146. In mathematics * Although composite, 145 is a Fermat pseudoprime in sixteen bases with b < 145. In four of those bases, it is a strong pseudoprime: 1, 12, 17, and 144. * the Mertens function returns 0. * 145 is a and a centered square number. * . 145 is the fourth number that is the sum of two different pairs of < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
113 (number)
113 (one hundred ndthirteen) is the natural number following 112 and preceding 114. Mathematics * 113 is the 30th prime number (following 109 and preceding 127), so it can only be divided by one and itself. 113 is a Sophie Germain prime, an emirp, an isolated prime, a Chen prime and a Proth prime as it is a prime number of the form 7\times 2^+1. 113 is also an Eisenstein prime with no imaginary part and real part of the form 3n - 1. In decimal, this prime is a primeval number and a permutable prime with 131 and 311. *113 is a highly cototient number and a centered square number. *113 is the denominator of 355/113, an accurate approximation to . Other uses *113 is also the atomic number of nihonium. * A113 is a Pixar recurring inside joke or Easter Egg, e.g.: (WALL-E ''WALL-E'' (stylized with an interpunct as ''WALL·E'') is a 2008 American animated Romance film, romantic science fiction film produced by Pixar Animation Studios for Walt Disney Pictures. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
85 (number)
85 (eighty-five) is the natural number following 84 (number), 84 and preceding 86 (number), 86. In mathematics 85 is: * the product of two prime numbers (5 and 17), and is therefore a semiprime of the form (5.q) where q is prime. * specifically, the 24th Semiprime, it being the fourth of the form (5.q). *together with 86 (number) , 86 and 87 (number), 87, forms the second cluster of three consecutive semiprimes; the first comprising 33 (number), 33, 34 (number), 34, 35 (number), 35. * with a prime aliquot sum of 23 (number), 23 in the short aliquot sequence (85,23 (number), 23,1 (number), 1,0). * an octahedral number. * a centered triangular number. * a centered square number. * a decagonal number. * the smallest number that can be expressed as a Fermat's theorem on sums of two squares, sum of two squares, with all squares greater than 1, in two ways, 85 = 92 + 22 = 72 + 62. * the length of the hypotenuse of four pythagorean triple, Pythagorean triangles. * a Smith number in deci ... [...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, and a twin prime with 59. As a centered square number, it is the sum of two consecutive squares, 5^2 + 6^2. It is also a centered decagonal number, and a centered hexagonal number. 61 is the fourth cuban prime of the form p = \frac where x = y + 1, and the fourth Pillai prime since 8! + 1 is divisible by 61, but 61 is not one more than a multiple of 8. It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ... 61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (1416) and 60 (1160). It is the sixth up/down or Euler zigzag number. 61 is the smallest ''proper prime'', a prime p which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length p - 1, where each digit (0, 1, ..., 9) ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
41 (number)
41 (forty-one) is the natural number following 40 (number), 40 and preceding 42 (number), 42. In mathematics 41 is: * the 13th smallest prime number. The next is 43 (number), 43, making both twin primes. * the sum of the first six prime numbers (2 + 3 + 5 + 7 + 11 + 13). * a regular prime * a Ramanujan prime * a harmonic prime * a good prime * the 12th Supersingular prime (moonshine theory), supersingular prime * a Newman–Shanks–Williams prime. * the smallest Sophie Germain prime to start a Cunningham chain of the first kind of three terms, . * an Eisenstein prime, with no imaginary part and real part of the form 3''n'' − 1. * a Proth prime as it is 5 × 23 + 1. * the smallest Lucky numbers of Euler, lucky number of Euler: the polynomial yields primes for all the integers ''k'' with . * the sum of two Square number, squares (42 + 52), which makes it a centered square number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
