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Beatty Sequence
In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926. Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number. Beatty sequences can also be used to generate Sturmian words. Definition Any irrational number r that is greater than one generates the Beatty sequence \mathcal_r = \lfloor r \rfloor, \lfloor 2r \rfloor, \lfloor 3r \rfloor,\ldots The two irrational numbers r and s = r/(r-1) naturally satisfy the equation 1/r + 1/s = 1. The two Beatty sequences \mathcal_r and \mathcal_s that they generate form a ''pair of complementary Beatty sequences''. Here, "complementary" means that every positive integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
11 (number)
11 (eleven) is the natural number following 10 (number), 10 and preceding 12 (number), 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables. Name "Eleven" derives from the Old English ', which is first attested in Bede's late 9th-century ''Ecclesiastical History of the English People''. It has cognates in every Germanic language (for example, German ), whose Proto-Germanic language, Proto-Germanic ancestor has been linguistic reconstruction, reconstructed as , from the prefix (adjectival "1 (number), one") and suffix , of uncertain meaning. It is sometimes compared with the Lithuanian language, Lithuanian ', though ' is used as the suffix for all numbers from 11 to 19 (analogously to "-teen"). The Old English form has closer cognates in Old Frisian, Old Saxon, Saxon, and Old Norse, Norse, whose ancestor has been reconstructed as . This was formerly thought to be derived from Proto-Germanic ("10 (number), ten"); it is now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5 (number)
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has attained significance throughout history in part because typical humans have five digits on each hand. In mathematics 5 is the third smallest prime number, and the second super-prime. It is the first safe prime, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second Fermat prime and the third Mersenne prime exponent, as well as the third Catalan number, and the third Sophie Germain prime. Notably, 5 is equal to the sum of the ''only'' consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, ( 3, 5) and (5, 7). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2 (number)
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. Evolution Arabic digit The digit used in the modern Western world to represent the number 2 traces its roots back to the Indic Brahmic script, where "2" was written as two horizontal lines. The modern Chinese and Japanese languages (and Korean Hanja) still use this method. The Gupta script rotated the two lines 45 degrees, making them diagonal. The top line was sometimes also shortened and had its bottom end curve towards the center of the bottom line. In the Nagari script, the top line was written more like a curve connecting to the bottom line. In the Arabic Ghubar writing, the bottom line was completely vertical, and the digit looked like a dotless closing question mark. Restoring the bottom line to its original horizontal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
29 (number)
29 (twenty-nine) is the natural number following 28 and preceding 30. Mathematics * 29 is the tenth prime number, and the fourth primorial prime. * 29 forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. * 29 is the sum of three consecutive squares, 22 + 32 + 42. * 29 is a Lucas prime, a Pell prime, and a tetranacci number. * 29 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 29 is also the 10th supersingular prime. * None of the first 29 natural numbers have more than two different prime factors. This is the longest such consecutive sequence. * 29 is a Markov number, appearing in the solutions to ''x'' + ''y'' + ''z'' = 3''xyz'': , , , , etc. * 29 is a Perrin number, preceded in the sequence by 12, 17, 22. * 29 is the smallest positive whole number that cannot be made from the numbers , using each exactly once and using only addition, subtraction, multiplication, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
27 (number)
27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28. In mathematics * Twenty-seven is a cube of 3: 3^3=3\times 3\times 3. 27 is also 23 (see tetration). There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra. 27 is also a decagonal number. * In decimal, it is the first composite number not divisible by any of its digits. * It is the radix (base) of the septemvigesimal positional numeral system. * 27 is the only positive integer that is 3 times the sum of its digits. * In a prime reciprocal magic square of the multiples of , the magic constant is 27. * In the Collatz conjecture (aka the "3n+1 conjecture"), a starting value of 27 requires 111 steps to reach 1, more than any number smaller than it. * The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional. * In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
25 (number)
25 (twenty-five) is the natural number following 24 and preceding 26. In mathematics It is a square number, being 52 = 5 × 5. It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g., 252 = 625; the other is 76. It is the smallest square that is also a sum of two (non-zero) squares: 25 = 32 + 42. Hence, it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the five consecutive single-digit odd natural numbers 1, 3, 5, 7, and 9. 25 is a centered octagonal number, a centered square number, a centered octahedral number, and an automorphic number. 25 percent (%) is equal to . It is the smallest decimal Friedman number as it can be expressed by its own digits: 52. It is also a Cullen number and a vertically symmetrical number. 25 is the smallest pseudoprime satisfying the congruence 7''n'' = 7 mod ''n''. 25 is the smallest aspiring number — a composite non- sociable nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
24 (number)
24 (twenty-four) is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta (Y), and for 10−24 (i.e., the reciprocal of 1024) yocto (y). These numbers are the largest and smallest number to receive an SI prefix to date. In mathematics 24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2''q'', where ''q'' is an odd prime. It is the smallest number with exactly eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors ( 36) is greater than itself, as well as a superabundant number. In number theory and algebra *24 is the smallest 5-hemiperfect number, as it has a half-integer abundancy index: *:1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = × 24 *24 is a semiperfect number, since adding up all the proper diviso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
22 (number)
22 (twenty-two) is the natural number following 21 and preceding 23. In mathematics 22 is a palindromic number and the eighth semiprime; its proper divisors are 1, 2, and 11. It is the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number. It is also a Perrin number, from a sum of 10 and 12. 22 is the fourth pentagonal number, the third hexagonal pyramidal number, and the third centered heptagonal number. The maximum number of regions into which five intersecting circles divide the plane is 22. 22 is also the quantity of pieces in a disc that can be created with six straight cuts, which makes 22 the seventh central polygonal number. \frac is a commonly used approximation of the irrational number , the ratio of the circumference of a circle to its diameter; where both 22 and 7 are consecutive hexagonal pyramidal numbers. 22 also features in another approximation for pi, here by Srinivasa Ramanujan from an approximate constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21 (number)
21 (twenty-one) is the natural number following 20 and preceding 22. The current century is the 21st century AD, under the Gregorian calendar. In mathematics 21 is: * a composite number, its proper divisors being 1, 3 and 7, and a deficient number as the sum of these divisors is less than the number itself. * a Fibonacci number as it is the sum of the preceding terms in the sequence, 8 and 13. * the fifth Motzkin number. * a triangular number, because it is the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21). * an octagonal number. * a Padovan number, preceded by the terms 9, 12, 16 (it is the sum of the first two of these) in the padovan sequence. * a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes. * the sum of the divisors of the first 5 positive integers (i.e., 1 + (1 + 2) + (1 + 3) + (1 + 2 + 4) + (1 + 5)) * the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19 (number)
19 (nineteen) is the natural number following 18 (number), 18 and preceding 20 (number), 20. It is a prime number. Mathematics 19 is the eighth prime number, and forms a sexy prime with 13 (number), 13, a twin prime with 17 (number), 17, and a cousin prime with 23 (number), 23. It is the third full reptend prime, the fifth Trinomial triangle#Central trinomial coefficients, 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 Waring's problem#The number g(k), g(k). * The sum of the squares of the first 19 primes is divisible by 19. *19 is the sixth Heegner number. 67 (number), 67 and 163 (number), 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of 9, nine total. * 19 is the third centered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
17 (number)
17 (seventeen) is the natural number following 16 (number), 16 and preceding 18 (number), 18. It is a prime number. Seventeen is the sum of the first four prime numbers. In mathematics 17 is the seventh prime number, which makes seventeen the fourth super-prime, as seven is itself prime. The next prime is 19 (number), 19, with which it forms a twin prime. It is a cousin prime with 13 (number), 13 and a sexy prime with 11 (number), 11 and 23 (number), 23. It is an emirp, and more specifically a permutable prime with 71 (number), 71, both of which are also supersingular prime (moonshine theory), supersingular primes. Seventeen is the sixth Mersenne prime exponent, yielding 131,071. Seventeen is the only prime number which is the sum of four consecutive primes: 2,3,5,7. Any other four consecutive primes summed would always produce an even number, thereby divisible by 2 and so not prime. Seventeen can be written in the form x^y + y^x and x^y - y^x, and, as such, it is a Leyland ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
