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#1 2020-07-18 17:49:48

ganesh
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Some special numbers

1. Perfect numbers: In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28. 28 = 1 + 2 + 4 + 7 + 14.

2. Amicable numbers: Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number.

The smallest pair of amicable numbers is (220, 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.)

3. Abundant number: An abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.

4. Deficient number: A number that is greater than the sum of all of its divisors except itself. The factors of 22 are 1, 2 and 11 and 22, and 1 + 2 + 11 = 14, which is less than 22, so 22 is a deficient number.

5. Happy Number: A happy number is defined by the following process:

Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).

6. Sad number: An unhappy number is a number that is not happy, i.e., a number such that iterating this sum-of-squared-digits map starting with. never reaches the number 1. The first few unhappy numbers are 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20, ...

7. Taxicab number: In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), also called the nth Hardy–Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta(2) =

The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy:

I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways."

8. Complex number: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation

Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) that is taken to satisfy the relation

so that solutions to equations like the preceding one can be found.

9. Transcendental number: In mathematics, a transcendental number is a number that is not algebraic - that is, not a root (i.e., solution) of a nonzero polynomial equation with integer or equivalently rational coefficients. The most popular transcendental numbers are

10. In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,


where the Greek letter phi
  or
represents the golden ratio. It is an irrational number that is a solution to the quadratic equation
, with a value of:


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#2 2020-07-27 06:21:43

pi_cubed
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Re: Some special numbers

11. Odious Numbers
In number theory, odious numbers are numbers that have an odd number of digits in their binary expansion. They determine the locations of the non-zero integers in the Thue-Morse sequence.

Some examples:

1 (1)
4 (100)
5 (101)
6 (110)
7 (111)

12. Evil Numbers
Evil numbers are the opposite of odious numbers. They have an even number of digits in their binary expansion and determine the locations of the zeroes in the Thue-Morse sequence.

Some examples:

2 (10)
3 (11)
8 (1000)
9 (1001)
10 (1010)


pi^3

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#3 2020-07-27 13:15:12

ganesh
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Re: Some special numbers

13. Triangular number : A triangular number or triangle number counts objects arranged in an equilateral triangle (thus triangular numbers are a type of figurate numbers, other examples being square numbers and cube numbers). The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers starting at the 0th triangular number, is

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...

The triangle numbers are given by the following explicit formulas:


where
is a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".

14. Mersenne prime: In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form

for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is
. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form
for some prime p.

The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ...  and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...

Numbers of the form

without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is

Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes.

As of July 2020, 51 Mersenne primes are known. The largest known prime number,

, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#4 2020-07-28 13:49:28

ganesh
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Re: Some special numbers

15) Liouville number

In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (p, q) with q > 1 such that

Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. They are precisely the transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#5 2021-05-14 15:01:21

ganesh
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Re: Some special numbers

16) Fibonacci Numbers

The Fibonacci Number Sequence was first presented in Leonardo Pisano's book, "Liber abaci" or "Book of Calculating". It is a sequence that I find to be very fascinating, and suprisingly it is a part of every day nature.

The Fibonacci sequence can be found in sea shell spirals, branching plants, petals on flowers, and in pine cones. I will explain this sequence to you in 3 different ways: the basic sequence, the rabbit problem, and the bees.

The Basic Sequence

The first twenty numbers of the sequence are as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.

The numbers are obtained by adding two numbers to get the next.

Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.

Fibonacci numbers are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly.

Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.

They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a pine cone's bracts.

Fibonacci sequences appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone, and the family tree of honeybees. Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers. Field daisies most often have petals in counts of Fibonacci numbers. In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series.

A Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#6 2021-05-14 15:09:04

ganesh
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Re: Some special numbers

17) Tribonacci Numbers

The tribonacci series is a generalization of the Fibonacci sequence where each term is the sum of the three preceding terms.

The Tribonacci Sequence:

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757,
4700770, 8646064, 15902591, 29249425, 53798080, 98950096, 181997601, 334745777, 615693474, 1132436852… so on

General Form of Tribonacci number:

a(n) = a(n-1) + a(n-2) + a(n-3)
with
a(0) = a(1) = 0, a(2) = 1.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#7 2021-05-14 15:27:35

ganesh
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Re: Some special numbers

18) Lucas number

The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.

The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ratio. The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between.

The first few Lucas numbers are

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#8 2021-05-15 02:54:40

ganesh
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Re: Some special numbers

19) Shannon number

The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of :

, based on an average of about
possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves.

Shannon's calculation

Shannon showed a calculation for the lower bound of the game-tree complexity of chess, resulting in about

possible games, to demonstrate the impracticality of solving chess by brute force, in his 1950 paper "Programming a Computer for Playing Chess". (This influential paper introduced the field of computer chess.)


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#9 2021-05-15 03:17:36

ganesh
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Re: Some special numbers

20) Graham's number

Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is named after mathematician Ronald Graham, who used the number in conversations with popular science writer Martin Gardner as a simplified explanation of the upper bounds of the problem he was working on. In 1977, Gardner described the number in Scientific American, introducing it to the general public. At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof. The number was described in the 1980 Guinness Book of World Records, adding to its popular interest. Other specific integers (such as TREE(3)) known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example in connection with Harvey Friedman's various finite forms of Kruskal's theorem. Additionally, smaller upper bounds on the Ramsey theory problem from which Graham's number derived have since been proven to be valid.

Graham's number is much larger than many other large numbers such as Skewes' number and Moser's number, both of which are in turn much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Thus Graham's number cannot be expressed even by physical universe-scale power towers of the form

.

However, Graham's number can be explicitly given by computable recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Graham. As there is a recursive formula to define it, it is much smaller than typical busy beaver numbers. Though too large to be computed in full, the sequence of digits of Graham's number can be computed explicitly through simple algorithms. The last 12 digits are ...262464195387. With Knuth's up-arrow notation, Graham's number is

, where


or


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#10 2021-05-15 15:00:44

ganesh
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Re: Some special numbers

21) Moser's number

Moser's number is the number represented by "2 in a megagon". Megagon is here the name of a polygon with "mega" sides (not to be confused with the polygon with one million sides).

Alternative notations:

use the functions square(x) and triangle(x)
let M(n, m, p) be the number represented by the number n in m nested p-sided polygons; then the rules are:


and

Moser's number

It has been proven that in Conway chained arrow notation,


and, in Knuth's up-arrow notation,

Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#11 2021-05-22 03:14:50

ganesh
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Re: Some special numbers

22) Skewes's number

In number theory, Skewes' number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number

for which

where π is the prime-counting function and li is the logarithmic integral function. Skewes' number is much larger, but it is now known that there is a crossing near

. It is not known whether it is the smallest.

Skewes' numbers

John Edensor Littlewood, who was Skewes' research supervisor, had proved in Littlewood (1914) that there is such a number (and so, a first such number); and indeed found that the sign of the difference

changes infinitely many times. All numerical evidence then available seemed to suggest that
was always less than
. Littlewood's proof did not, however, exhibit a concrete such number
.

Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number

violating
below

.

In Skewes (1955), without assuming the Riemann hypothesis, Skewes proved that there must exist a value of

below

.

Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to Georg Kreisel, this was at the time not considered obvious even in principle.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#12 2021-05-22 17:16:58

ganesh
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Re: Some special numbers

23) Euler's number

The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of the natural logarithm. It is the limit of

as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

It is also the unique positive number a such that the graph of the function

has a slope of 1 at x = 0.

The (natural) exponential function

is the unique function which is equal to its own derivative, with the initial value f(0) = 1 (and hence one may define e as f(1)). The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one. There are various other characterizations.

e is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler (not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant), or Napier's constant. However, Euler's choice of the symbol e is said to have been retained in his honor. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number e has eminent importance in mathematics, alongside 0, 1,

, and i. All five of these numbers play important and recurring roles across mathematics, and these five constants appear in one formulation of Euler's identity. Like the constant
, e is irrational (that is, it cannot be represented as a ratio of integers) and transcendental (that is, it is not a root of any non-zero polynomial with rational coefficients). To 50 decimal places the value of e is:

2.71828182845904523536028747135266249775724709369995....


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#13 2021-05-24 14:38:18

ganesh
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Re: Some special numbers

24) Euler numbers

In mathematics, the Euler numbers are a sequence

of integers defined by the Taylor series expansion

,
where cosh t is the hyperbolic cosine. The Euler numbers are related to a special value of the Euler polynomials, namely:

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#14 2021-06-05 14:19:59

ganesh
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Re: Some special numbers

25) Rayo's number

Rayo's number is one of the largest named numbers, coined in a large number battle pitting Agustín Rayo against Adam Elga. Rayo's number is, in Rayo's own words, "the smallest positive integer bigger than any finite positive integer named by an expression in the language of first-order set theory with googol symbols or less."

By letting the number of symbols range over the natural numbers, we get a very quickly growing function Rayo(n). Rayo's number is Rayo(

). Rayo's function is uncomputable, which means that it is impossible for a Turing machine (and, by the Church–Turing thesis, any modern computer) to calculate Rayo(n). Indeed, if f is a function definable in a first order segment of the second order set theory assumed in the definition of Rayo's function, the defining formula Φ(n,m) of the predicate m=f(n) gives a lower bound of the composition of Rayo(n) and a sufficiently slow growing function depending on the growth rate of the length of
regarded as a function on n, where ┌n┐ is a fixed formalisation of n in the language of first order set theory.

Although the second order set theory was unspecified in the original definition and is clarified as the philosophic (but mathematically ill-defined) collection of formulae which the real world philosophically "satisfy", it is reasonable to assume that ZFC set theory is a first order segment of the unspecified set theory because the majority of mathematicians and googologists are interested in ZFC set theory. Under the assumption, Rayo's function outgrows all functions definable in ZFC set theory. Throughout this article, we always use the same assumption except for Axiom section which more deeply explains the issue on the lack of the clarification of the second order set theory.

Rayo's function is one of the most fast-growing functions ever to arise in professional mathematics; only a few functions, especially its extension, Fish number 7 surpasses it. Since Rayo's function uses difficult mathematics, there are several trials to generalise it which result in failure. For example, the FOOT (first-order oodle theory) function was also considered to surpass it, but it is ill-defined.

Rayo's function naturally can be outgrown by

for some countable
in the fast-growing hierarchy equipped with a fixed system of fundamental sequences for ordinals
. Indeed, it is outgrown by
with respect to the fundamental sequence ω[n]=Rayo(n). On the other hand, it can never be outgrown by
for a countable ordinal
and a fixed system of fundamental sequences for ordinals
if the hierarchy is defined in ZFC set theory by the definition of Rayo.

(Zermelo–Fraenkel set theory is a first-order axiomatic set theory. Under this name are known two axiomatic systems - a system without axiom of choice (abbreviated ZF) and one with axiom of choice (abbreviated ZFC). Both systems are very well known foundational systems for mathematics, thanks to their expressive power.

Although different axiomatizations of set theory are possible, ZF and ZFC are the most common and well-known ones.)


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#15 2021-06-10 16:54:00

ganesh
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Re: Some special numbers

26) Twin prime

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.

Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved.

Properties

Usually the pair (2, 3) is not considered to be a pair of twin primes. Since 2 is the only even prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes.

The first few twin prime pairs are:

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), …

Five is the only prime that belongs to two pairs, as every twin prime pair greater than

is of the form
for some natural number n; that is, the number between the two primes is a multiple of 6. As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#16 2021-06-28 01:28:49

ganesh
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Re: Some special numbers

27) Superior highly composite number

In mathematics, a superior highly composite number is a natural number which has more divisors than any other number scaled relative to some positive power of the number itself. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.

For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k smaller than n we have


and for all natural numbers k larger than n we have


where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan (1915).

The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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