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Ruth–Aaron Pair
In mathematics, a Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime factors of each integer are equal: :714 = 2 × 3 × 7 × 17, :715 = 5 × 11 × 13, and : 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29. There are different variations in the definition, depending on how many times to count primes that appear multiple times in a factorization. The name was given by Carl Pomerance for Babe Ruth and Hank Aaron, as Ruth's career regular-season home run total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Pomerance was a mathematician at the University of Georgia at the time Aaron (a member of the nearby Atlanta Braves) broke Ruth's record, and the student of one of Pomerance's colleagues noticed that the sums of the prime factors of 714 and 715 were equal. Examples If only distinct prime factors are counted, the first few Ruth–Aaron pairs are: :( 5, 6), ( 24, 25), ( 49, 50), ( 77, ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
49 (number)
49 (forty-nine) is the natural number following 48 (number), 48 and preceding 50 (number), 50. In mathematics Forty-nine is the square of 7, seven. 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 (number), 100 not having its home prime known (). Decimal representation The sum of the digits of the square of 49 (2401) is the square root of 49. 49 is the first square where the digits are squares. In this case, 4 and 9 are squares. Reciprocal The fraction is a repeating decimal with a period of 42: : = (42 digits repeat) There are 42 (note that this number is the period) positive integers that are less than 49 and coprime to 49. Multiplying 020408163265306122448979591836734693877551 by each of these integers results in a cyclic permutation of the original number: *020408163265306122448979591836734693877551 × 2 = 040 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
126 (number)
126 (one hundred ndtwenty-six) is the natural number following 125 and preceding 127. In mathematics As the binomial coefficient \tbinom, 126 is a central binomial coefficient and a pentatope number. It is also a decagonal number, a Harshad number and a pentagonal pyramidal number. As 125 + 1 it is σ3(5), the fifth value of the sum of cubed divisors function, and is a sum of two cubes. There are exactly 126 crossing points among the diagonals of a regular nonagon, 126 binary strings of length seven that are not repetitions of a shorter string, 126 different semigroups on four elements (up to isomorphism and reversal), and 126 different ways to partition a decagon into even polygons by diagonals. There are exactly 126 positive integers that are not solutions of the equation :x=abc+abd+acd+bcd, where ''a'', ''b'', ''c'', and ''d'' must themselves all be positive integers. It is the fifth Granville number, and the third such not to be a perfect number. Also, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
125 (number)
125 (one hundred ndtwenty-five) is the natural number following 124 and preceding 126. In mathematics 125 is the cube of 5. It can be expressed as a sum of two squares in two different ways, 125 = 10² + 5² = 11² + 2². 125 and 126 form a Ruth-Aaron pair under the second definition in which repeated prime factors are counted as often as they occur. Like many other powers of 5, it is a Friedman number in base 10 since 125 = 51 + 2. 125 is the center of a close triplet of perfect powers, (121 = 112, 125 = 53, 128 = 27). Excluding the trivial cases of 0 and 1, the only closer such triplet is (4,8,9) and the only other equally close is (25, 27, 32). U.S. military * Air National Guard 125th Special Tactics Squadron unit in Portland, Oregon * US Air Force 125th Fighter Wing, Air National Guard unit at Jacksonville International Airport, Florida * US Navy VAW-125 squadron at Naval Station Norfolk, Virginia * US Navy VFA-125 strike fighter squadron at Naval Air Station Lemoore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
16 (number)
16 (sixteen) is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and . In English speech, the numbers 16 and 60 are sometimes confused, as they sound very similar. Sixteen is the fourth power of two. For this reason, 16 was used in weighing light objects in several cultures. The British have 16 ounces in one pound; the Chinese used to have 16 ''liangs'' in one ''jin''. In old days, weighing was done with a beam balance to make equal splits. It would be easier to split a heap of grains into sixteen equal parts through successive divisions than to split into ten parts. Chinese Taoists did finger computation on the trigrams and hexagrams by counting the finger tips and joints of the fingers with the tip of the thumb. Each hand can count up to 16 in such manner. The Chinese abacus uses two upper beads to represent the 5s and 5 low ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
15 (number)
15 (fifteen) is the natural number following 14 and preceding 16. Mathematics 15 is: * A composite number, and the sixth semiprime; its proper divisors being , and . * A deficient number, a smooth number, a lucky number, a pernicious number, a bell number (i.e., the number of partitions for a set of size 4), a pentatope number, and a repdigit in binary (1111) and quaternary (33). In hexadecimal, and higher bases, it is represented as F. * A triangular number, a hexagonal number, and a centered tetrahedral number. * The number of partitions of 7. * The smallest number that can be factorized using Shor's quantum algorithm. * The magic constant of the unique order-3 normal magic square. * The number of supersingular primes. Furthermore, * 15 is one of two numbers within the ''teen'' numerical range (13-19) not to use a single-digit number in the prefix of its name (the first syllable preceding the ''teen'' suffix); instead, it uses the adjective form of five (' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9 (number)
9 (nine) is the natural number following and preceding . Evolution of the Arabic digit In the beginning, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a -look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ encircles a lowercase ''a''. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic. While the shape of the glyph for the digit 9 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in . The mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8 (number)
8 (eight) is the natural number following 7 and preceding 9. In mathematics 8 is: * a composite number, its proper divisors being , , and . It is twice 4 or four times 2. * a power of two, being 2 (two cubed), and is the first number of the form , being an integer greater than 1. * the first number which is neither prime nor semiprime. * the base of the octal number system, which is mostly used with computers. In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an wikt:octet, octet. * a Fibonacci number, being plus . The next Fibonacci number is . 8 is the only positive Fibonacci number, aside from 1, that is a perfect cube. * the only nonzero perfect power that is one less than another perfect power, by Catalan conjecture, Mihăilescu's Theorem. * the order of the smallest non-abelian group all of whose subgroups are normal. * the dimension of the octonions and is the highest possible dimension of a normed divisio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
154 (number)
154 (one hundred ndfifty-four) is the natural number following 153 and preceding 155. In mathematics 154 is a nonagonal number. Its factorization makes 154 a sphenic number There is no integer with exactly 154 coprimes below it, making 154 a noncototient, nor is there, in base 10, any integer that added up to its own digits yields 154, making 154 a self number 154 is the sum of the first six factorials, if one starts with 0! and assumes that 0!=1. With just 17 cuts, a pancake can be cut up into 154 pieces (Lazy caterer's sequence). The distinct prime factors of 154 add up to 20, and so do the ones of 153, hence the two form a Ruth-Aaron pair. 154! + 1 is a factorial prime. In music * 154 is an album by Wire, named for the number of live gigs Wire had performed at that time In the military * was a United States Navy ''Trefoil''-class concrete barge during World War II * was a United States Navy ''Admirable''-class minesweeper during World War II * was a United States Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
153 (number)
153 (one hundred ndfifty-three) is the natural number following 152 and preceding 154. In mathematics The number 153 is associated with the geometric shape known as the Vesica Piscis or Mandorla. Archimedes, in his ''Measurement of a Circle'', referred to this ratio (153/265), as constituting the "measure of the fish", this ratio being an imperfect representation of 1/. As a triangular number, 153 is the sum of the first 17 integers, and is also the sum of the first five positive factorials:1!+2!+3!+4!+5!.Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987): 140–141. The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers. The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair. Since 153 = 1^3 + 5^3 + 3^3, it is a 3-narcissistic number, and it is also the smallest three-digit number which ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
105 (number)
105 (one hundred ndfive) is the natural number following 104 and preceding 106. In mathematics 105 is a triangular number, a dodecagonal number, and the first Zeisel number. It is the first odd sphenic number and is the product of three consecutive prime numbers. 105 is the double factorial of 7. It is also the sum of the first five square pyramidal numbers. 105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such numbers less than a thousand are 9, 15, 195, and 825. 105 is also the middle of the only prime sextuplet (97, 101, 103, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the first three odd primes (3\times5\times7) and less than the square of the next prime (11) by > 8, for n=105, n ± 2, ± 4, and ± 8 must be prime, and n ± 6, ± 10, ± 12, and ± 14 must be composite (prime gap). 105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83, and 97. The distinct prime factors of 105 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
104 (number)
104 (one hundred ndfour) is the natural number following 103 and preceding 105. In mathematics 104 is a primitive semiperfect number and a composite number, with its divisors being 1, 2, 4, 8, 13, 26, 52 and 104. As it has 8 divisors total, and 8 is one of those divisors, 104 is a refactorable number. The distinct prime factors of 104 add up to 15, and so do the ones of 105, hence the two numbers form a Ruth-Aaron pair under the first definition. In regular geometry, 104 is the smallest number of unit line segments that can exist in a plane with four of them touching at every vertex.A figure made up of a row of 4 adjacent congruent rectangles is divided into 104 regions upon drawing diagonals of all possible rectangles φ(104) = φ(σ(104)). In science *The atomic number of rutherfordium. *Number of degrees Fahrenheit corresponding to 40 Celsius. In other fields 104 is also: *The number of Corinthian columns in the Temple of Olympian Zeus, the largest temple ever built ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
