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Centered Square Number
In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a 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 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 square, 9 (sum of two blue triangles): Relationships with o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Polygonal 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] |
Centered Polygonal 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] |
Elementary Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects i ... [...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 learner's permit used to be called "365" because it was valid for only 366 days. Financial and scientific calculations often use a 365-day calendar to simplify daily rates. Religious meanings Judeo-Christian In the Jewish faith there are 365 " negative commandments". Also, the Bibl ... [...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 prime number *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. In religion * The number of soldiers that Muhammad had with him in the first battle fought by the Muslims, the Battle of Badr. * In Twelver Shia Islam, 313 is the number of soldiers or generals that will be in the army of the 12th "Imam of time" (Mahdi). In other fields *The number 313 is the U.S. telephone area code for the city of Detroit and nearby locales. *Frame 313 of the Zapruder film shows the moment of impact for the bullet that ki ... [...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. In other fields In Texas hold 'em, the probability of being dealt pocket aces (the strongest possible outcome in the initial deal of two cards per player) is 1/221. Sherlock Holmes's home address: 221B Baker Street 221B Baker Street is the London address of the fictional detective Sherlock Holmes, created by author Sir Arthur Conan Doyle. In the United Kingdom, postal addresses with a number followed by a letter may indicate a separate address within .... References {{DEFAULTSORT:221 (Number) Integers ... [...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 an odd number * 181 is a centered number ** 181 is a centered pentagonal number ** 181 is a centered 12-gonal number ** 181 is a centered 18-gonal number ** 181 is a centered 30-gonal number ** 181 is a centered square number ** 181 is a star number that represents a centered hexagram (as in the game of Chinese checkers) * 181 is a deficient number, as 1 is less than 181 * 181 is an odious number * 181 is a prime number ** 181 is a Chen prime ** 181 is a dihedral prime ** 181 is a full reptend prime ** 181 is a palindromic prime ** 181 is a strobogrammatic prime, the same when viewed upside down ** 181 is a twin prime with 179 * 181 is a square-free number * 181 is an undulating number, if written in the ternary, the negaternary, or the nonary numeral systems * 181 is the difference of 2 square numbers: 912 – 902 * 181 is the sum of 2 consecutive square numbers: ... [...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 to sixteen bases with b < 145. In four of those bases, it is a : 1, 12, 17, and 144. * Given 145, the returns 0. * 145 is a and a . * |
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 × 24 + 1. 113 is also an Eisenstein prime with no imaginary part and real part of the form 3n - 1. In base 10, 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 . See also * 113 (other) * A113 is A Pixar recurring inside joke or Easter Egg, e.g.: (WALL-E) = (W-A113). References * Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers ''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
85 (number)
85 (eighty-five) is the natural number following 84 and preceding 86. In mathematics 85 is: * the product of two prime numbers (5 and 17), and is therefore a semiprime; specifically, the 24th biprime not counting perfect squares. Together with 86 and 87, it forms the second cluster of three consecutive biprimes. * an octahedral number. * a centered triangular number. * a centered square number. * a decagonal number. * the smallest number that can be expressed as a 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 triangles. * a Smith number in decimal. In astronomy * Messier object M85 is a magnitude 10.5 lenticular galaxy in the constellation Coma Berenices * NGC 85 is a galaxy in the constellation Andromeda * 85 Io is a large main belt asteroid * 85 Pegasi is a multiple star system in constellation of Pegasus * 85 Ceti is a variable star in the constellation of Cetus * 85D/Boethin i ... [...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] |
41 (number)
41 (forty-one, XLI) is the natural number following 40 and preceding 42. In mathematics * the 13th smallest prime number. The next is 43, making both twin primes. * the sum of the first six prime numbers (2 + 3 + 5 + 7 + 11 + 13). * the 12th 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 largest lucky number of Euler: the polynomial yields primes for all the integers ''k'' with . * the sum of two squares, 42 + 52. * the sum of the sum of the divisors of the first 7 positive integers. * the smallest integer whose reciprocal has a 5-digit repetend. That is a consequence of the fact that 41 is a factor of 99999. * the smallest integer whose s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
