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Lucky Number
In number theory, a lucky number is a natural number in a set which is generated by a certain " sieve". This sieve is similar to the sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers). The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. In the same work they also suggested calling another sieve, "the sieve of Josephus Flavius" because of its similarity with the counting-out game in the Josephus problem. Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and twin primes also appear to occur with similar frequency. However, if ''L''''n'' denotes the ''n''-th lucky number, and ''p''''n'' the ''n''-th prime, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Complex analysis, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. In mathematics The number 1 is the first natural number after 0. Each natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
51 (number)
51 (fifty-one) is the natural number following 50 and preceding 52. In mathematics Fifty-one is * a pentagonal number as well as a centered pentagonal number and an 18-gonal number *the 6th Motzkin number, telling the number of ways to draw non-intersecting chords between any six points on a circle's boundary, no matter where the points may be located on the boundary. *a Perrin number, coming after 22, 29, 39 in the sequence (and the sum of the first two) * a Størmer number, since the greatest prime factor of 512 + 1 = 2602 is 1301, which is substantially more than 51 twice. *There are 51 different cyclic Gilbreath permutations on 10 elements, and therefore there are 51 different real periodic points of order 10 on the Mandelbrot set.. *Since 51 is the product of the distinct Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
49 (number)
49 (forty-nine) is the natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ... following 48 and preceding 50. In mathematics Forty-nine is the square of the prime number seven and hence the fourth non-unitary square prime of the form ''p''2. Both of its digits are square numbers, 4 being the square of 2 and 9 being the square of 3. It appears in the Padovan sequence, preceded by the terms 21, 28, 37 (it is the sum of the first two of these). Along with the number that immediately derives from it, 77, the only number under 100 not having its home prime known (). The smallest triple of three squares in arithmetic succession is (1,25,49), and the second smallest is (49,169,289). 49 is the smallest discriminant of a totally real cubic field. 49 and 94 are the onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
43 (number)
43 (forty-three) is the natural number following 42 (number), 42 and preceding 44 (number), 44. Mathematics 43 is a prime number, and a twin prime of 41 (number), 41. 43 is the smallest prime that is not a Chen prime. 43 is also a Wagstaff prime, and a Heegner number. 43 is the fourth term of Sylvester's sequence. 43 is the largest prime which divides the order of the Janko group J4, Janko group J4. Netherlands, Dutch mathematician Hendrik Lenstra wrote a mathematical research paper discussing the properties of the number, titled ''Ode to the number 43.'' Notes Further reading Hendrik Lenstra, Lenstra, Hendrik (2009)''Ode to the number 43'' (In Dutch). Nieuw Archief voor Wiskunde, Nieuw Arch. Wiskd. Amsterdam, NL: Koninklijk Wiskundig Genootschap (5) 10, No. 4: 240-244. {{Integers, zero Integers ... [...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 3rd isolated prime without a twin prime. 37 is the first irregular prime with irregularity index of 1, where the smallest prime number with an irregularity index of 2 is the thirty-seventh prime number, 157. The smallest magic square, using only primes and 1, contains 37 as the value of its central cell: Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11). 37 requires twenty-one steps to return to 1 in the Collatz problem, as do adjacent numbers 36 and 38. The two closest numbers to cycle through the elementary Collatz pathway are 5 and 32, whose sum is 37; also, the trajectories for 3 and 21 both require seven steps to reach 1. On the other hand, the first two integers that return 0 for the Mertens function ( 2 and 39) have a difference of 37, where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
33 (number)
33 (thirty-three) is the natural number following 32 (number), 32 and preceding 34 (number), 34. In mathematics 33 is the 21st composite number, and 8th distinct semiprime (third of the form 3 \times q where q is a higher prime). It is one of two numbers to have an aliquot sum of 15 (number), 15 = 3 × 5 — the other being the Square number, square of 4 — and part of the aliquot sequence of 9 = 32 in the aliquot tree (33, 15 (number), 15, 9, 4 (number), 4, 3 (number), 3, 2, 1). It is the largest positive integer that cannot be expressed as a sum of different triangular numbers, and it is the largest of twelve integers that are not the sum of five non-zero squares; on the other hand, the 33rd triangular number 561 (number), 561 is the first Carmichael number. 33 is also the first non-trivial dodecagonal number (like 369, and 561) and the first non-unitary centered dodecahedral number. It is also the sum of the first four positive factorials, and the sum of the sums of the divi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
31 (number)
31 (thirty-one) is the natural number following thirty, 30 and preceding 32 (number), 32. It is a prime number. Mathematics 31 is the 11th prime number. It is a superprime and a Self number#Self primes, self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is the third Mersenne prime of the form 2''n'' − 1, and the eighth Mersenne prime ''exponent'', in-turn yielding the maximum positive value for a 32-bit Integer (computer science), signed binary integer in computing: 2,147,483,647. After 3, it is the second Mersenne prime not to be a double Mersenne prime, while the 31st prime number (127 (number), 127) is the second double Mersenne prime, following 7. On the other hand, the thirty-first triangular number is the perfect number 496 (number), 496, of the form 2(5 − 1)(25 − 1) by the Euclid-Euler theorem. 31 is also a ''primorial prime'' like its twin prime (29 (number), 29), as well as both a lucky prime and a happy number like its d ... [...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, and hence the third non-unitary square prime of the form ''p''2. 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. 25 has an even aliquot sum of 6, which is itself the first even and perfect number root of an aliquot sequence; not ending in ( 1 and 0). 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 ... [...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. Mathematics Twenty-one is the fifth distinct semiprime, and the second of the form 3 \times q where q is a higher prime. It is a repdigit in quaternary (1114). Properties As a biprime with proper divisors 1, 3 and 7, twenty-one has a prime aliquot sum of 11 within an aliquot sequence containing only one composite number (21, 11, 1, 0). 21 is the first member of the second cluster of consecutive discrete semiprimes (21, 22), where the next such cluster is ( 33, 34, 35). There are 21 prime numbers with 2 digits. There are a total of 21 prime numbers between 100 and 200. 21 is the first Blum integer, since it is a semiprime with both its prime factors being Gaussian primes. While 21 is the sixth triangular number, it is also the sum of the divisors of the first five positive integers: \begin 1 & + 2 + 3 + 4 + 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
15 (number)
15 (fifteen) is the natural number following 14 (number), 14 and preceding 16 (number), 16. Mathematics 15 is: * The eighth composite number and the sixth semiprime and the first odd and fourth discrete semiprime; its proper divisors are , , and , so the first of the form (3.q), where q is a higher prime. * a deficient number, a lucky number, a bell number (i.e., the number of partitions for a set of size 4), a pentatope number, and a repdigit in Binary numeral system, binary (1111) and quaternary numeral system, quaternary (33). In hexadecimal, and higher bases, it is represented as F. * with an aliquot sum of 9 (number), 9; within an aliquot sequence of three composite numbers (15,9 (number), 9,4 (number), 4,3 (number), 3,1 (number), 1,0) to the Prime in the 3 (number), 3-aliquot tree. * the second member of the first cluster of two discrete semiprimes (14 (number), 14, 15); the next such cluster is (21 (number), 21, 22 (number), 22). * the first number to be Polygonal numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
13 (number)
13 (thirteen) is the natural number following 12 (number), 12 and preceding 14 (number), 14. Folklore surrounding the number 13 appears in many cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars (there are approximately 12.41 lunations per solar year, and hence 12 "true months" plus a smaller, and often portentous, thirteenth month). This can be witnessed, for example, in the "Twelve Days of Christmas" of Western European tradition. In mathematics The number 13 is a prime number, happy number and a lucky number. It is a twin prime with 11 (number), 11, as well as a cousin prime with 17 (number), 17. It is the second of only 3 Wilson prime, Wilson primes: 5, 13, and 563 (number), 563. A 13-sided regular polygon is called a tridecagon. List of basic calculations In languages Grammar * In all Germanic languages, 13 is the first Compound (linguistics), compound number; the numbers 11 and 12 have their own names. * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
