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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)


pi³

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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. 

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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. 

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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. 

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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. 

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#17 2021-10-19 15:29:54

ganesh
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Re: Some special numbers

28) Some Ginormous Numbers


Graham's number,






PS: Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g.

.

As with most combinatorial notations, the definition is recursive. In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power.

Graham's number

itself cannot be expressed concisely in Conway chained arrow notation, but it is bounded by the following:


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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#18 2021-10-27 19:16:01

ganesh
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Re: Some special numbers

29) Belphegor's prime number

Belphegor's prime number is a palindromic prime number with a 666 hiding between 13 zeros and a 1 on either side. The ominous number can be abbreviated as
1 0(13) 666 0(13) 1, where the (13) denotes the number of zeros between the 1 and 666.

Although he didn't "discover" the number, scientist and author Cliff Pickover made the sinister-feeling number famous when he named it after Belphegor (or Beelphegor), one of the seven demon princes of hell.

The number apparently even has its own devilish symbol, which looks like an upside-down symbol for pi. According to Pickover's website, the symbol is derived from a glyph in the mysterious Voynich manuscript, an early 15th-century compilation of illustrations and text that no one seems to understand.


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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#19 2021-10-27 19:55:26

ganesh
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Registered: 2005-06-28
Posts: 34,872

Re: Some special numbers

30) Tau

You know what's cooler than ONE pie? … TWO pies. In other words, two times pi, or the number "tau," which is roughly 6.28.

"Using tau makes every formula clearer and more logical than using pi," said John Baez, a mathematician at the University of California, Riverside. "Our focus on pi rather than 2pi is a historical accident."

Tau is what shows up in the most important formulas, he said.

While pi relates a circle's circumference to its diameter, tau relates a circle's circumference to its radius — and many mathematicians argue that this relationship is much more important. Tau also makes seemingly unrelated equations nicely symmetrical, such as the one for a circle's area and an equation describing kinetic and elastic energy.

Tau – A Mathematical Constant

What is Tau?

The constant is numerically equal to 2*pi (2 times pi), and with value approximately 6.28. The ratio equates to 2*C/D. Where C is circumference and D is diameter of circle.

Applications of Tau

* There are many expressions that actually require “2*pi” calculation, having tau being equal to that simplifies them to great extent, for e.g Circumference of circle = 2*pi*r = tau*r.
* Concept of tau can be useful in angular measurements like angles in radians, representing as a complete “one-turn” and cos,sine functions in trigonometry have period of tau.
* These concepts can be useful for teaching geometry as would reduce the confusion of using “pi” and “2*pi” at many applications and would help get rid of factor of 2.
* Tau simplifies euler’s identity by eradicating the factor of 2.
* It is useful at many places where “2*pi” are used such as fourier transforms, cauchy integral formula’s etc.

Criticism against Tau

* Since it contradicts with the symbols of torque, shear stress and time, this symbol has been a lot of criticism.
* We already had a ratio of “C/D” equal to pi, having another circle ratio with factor of two will create confusion in choice.
* There exist formulas which look more elegant as expression of “pi” rather than tau, for example, area of circle = pi*r*r = (tau*r*r)/2, introducing an extra factor of “1/2”.

Coding Prospects

Since Programming has always been trying to match up with mathematical advancements, symbol of tau has been introduced as a constant in recent python 3.6 under the math module.

Output: 

The value of tau (using 2*pi) is : 6.283185307179586
The value of tau (using in-built tau) is : 6.283185307179586.

Let's Use Tau--It's Easier Than Pi

There aren't many things that Congress can agree on, but in early 2009 it passed a bipartisan resolution designating March 14th of each year as "Pi Day." Pi, the mathematical constant that students first encounter with the geometry of circles, equals about 3.14, hence its celebration on March 14. The math holiday had been a staple of geeks and teachers for years—festivities include eating pie the pastry while talking about pi the number—but dissent began to appear from an unexpected quarter: a vocal and growing minority of mathematicians who rally around the radical proposition that pi is wrong.

They don't mean anything has been miscalculated. Pi (π) still equals the same infinite string of never-repeating digits. Rather, according to The Tau Manifesto, "pi is a confusing and unnatural choice for the circle constant." Far more relevant, according to the algebraic apostates, is 2π, aka tau.

Manifesto author Michael Hartl received his PhD in theoretical physics from the California Institute of Technology and is only one in a string of established players beginning to question the orthodoxy. Last year the University of Oxford hosted a daylong conference titled "Tau versus Pi: Fixing a 250-Year-Old Mistake." In 2012 the Massachusetts Institute of Technology modified its practice of letting applicants know admissions decisions on Pi Day by further specifying that it will happen at tau time—that is, at 6:28 P.M. The Internet glommed onto the topic as well, with its traditional fervor for whimsical causes. YouTube videos on the subject abound with millions of views and feisty comment sections—hardly a common occurrence in mathematical debates.

The crux of the argument is that pi is a ratio comparing a circle’s circumference with its diameter, which is not a quantity mathematicians generally care about. In fact, almost every mathematical equation about circles is written in terms of r for radius. Tau is precisely the number that connects a circumference to that quantity.

But usage of pi extends far beyond the geometry of circles. Critical mathematical applications such as Fourier transforms, Riemann zeta functions, Gaussian distributions, roots of unity, integrating over polar coordinates and pretty much anything involving trigonometry employs pi. And throughout these diverse mathematical areas the constant π is preceded by the number 2 more often than not. Tauists (yes, they call themselves tauists) have compiled exhaustively long lists of equations—both common and esoteric, in both mathematics and physics—with 2π holding a central place. If 2π is the perennial theme, the almost magically recurring number across myriad branches of mathematics, shouldn’t that be the fundamental constant we name and celebrate?

If that’s all there was, the tau movement would likely be a curiosity and nothing more. But reasons for switching to tau are deeply rooted in pedagogy as well. University of Utah mathematics professor Robert Palais, who is considered the founding father of the movement, started the "pi is wrong" ruckus with an article of the same name in 2001[pdf]. The article, which should be required reading for all advanced high school students, creates a tantalizing picture of how much easier certain fundamental concepts of trigonometry could be in an alternate universe where we use tau. For example, with pi-based thinking, if you want to designate a point one third of the way around the circle, you say it has gone two thirds pi radians. Three quarters around the same circle has gone one and a half pi radians. Everything is distorted by a confusing factor of two. By contrast, a third of a circle is a third of tau. Three quarters of a circle is three quarters tau. As a result of pi, Palais says, "the opportunity to impress students with a beautiful and natural simplification is turned into an absurd exercise in memorization and dogma."

At its heart, pi refers to a semicircle, whereas tau refers to the circle in its entirety. Mathematician and poet Mike Keith once wrote a 10,000 word poem dedicated to the first 10,000 digits of pi. He is now a proponent of tau. According to a PBS article from last year, he said that thinking in terms of pi is like reaching your destination and saying you're twice halfway there.

For mathematicians, pi obscures some of the underlying symmetries of mathematics and muddies up what should be elegant with extraneous factors of two. There’s an admittedly grandiose idea that mathematics is the language with which we express and see certain underpinning truths to the universe. To clutter that language with superfluous twos would be as bad as littering a Shakespearean monologue with “likes” and “umms” and “whatevers.” As the Bard nearly wrote, “Knowledge is two of the half-wings wherewith we fly to heaven.”

We Americans have almost a proud tradition of using poorly chosen units because of inertia: Fahrenheit instead of Celsius, miles instead of kilometers. Even the great Benjamin Franklin inadvertently established the convention of calling positive charge negative and vice-versa as a result of his experiments with electricity.

Indeed, the whole problem began as a historical accident, tauists say. In early civilizations a diameter was an easier quantity to measure than a radius. So when the Babylonians or Egyptians wanted rules of thumb for their architecture, a ratio of circumference to diameter is what they turned to. (The two civilizations estimated it to be 3.125 and 3.16, respectively.) Even the Bible specifies the ratio of a circle’s diameter to its circumference: “And [Hiram] made a molten sea, 10 cubits from the one brim to the other: it was round all about, and…a line of thirty cubits did compass it round about” (1 Kings 7:23).

The Greeks used formal geometric proofs to estimate the circumference-to-diameter ratio. Archimedes (he of the lever and shouts of "Eureka!") found strict lower and upper bounds of 3.1408 and 3.1429. Yet his choice of comparing the circumference with diameter was arbitrary; he could just as easily have used radius instead. (Interestingly, Archimedes did not use the Greek letter π. That didn't come until Swiss mathematician Leonhard Euler popularized the convention in 1736, and even he seemed to be ambivalent about whether to define π as 3.14 or as the 6.28 we now write as τ.)

Although switching to tau when all the textbooks and academic papers use pi may sound daunting, it doesn’t need to be. There could be a transitional period of using both mathematical constants while we phase out the old and humor the intransigents who can’t or won’t change.

Asked in an e-mail about the reaction his original piece has received, Palais is humbled. "I never would have imagined the scale of the discussion," he says. And given that it's already far exceeded his expectations, he expresses optimism that it could continue even further.


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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#20 2021-11-02 14:01:46

ganesh
Administrator
Registered: 2005-06-28
Posts: 34,872

Re: Some special numbers

31) Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is

, that is, e raised to the power
. Like both e and
, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that


where i is the imaginary unit. Since −i is algebraic but not rational,
is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is
, known as the Gelfond–Schneider constant. The related value
is also irrational.

The decimal expansion of Gelfond's constant begins

23.1406926327792690057290863679485473802661062426002119934450464095243423506904527835169719970675492196....


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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#21 2021-11-02 14:24:15

ganesh
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Registered: 2005-06-28
Posts: 34,872

Re: Some special numbers

32) Gelfond–Schneider constant

The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two:

= 2.6651441426902251886502972498731...

which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general Gelfond–Schneider theorem, which solved the part of Hilbert's seventh problem described below.

Properties

The square root of the Gelfond–Schneider constant is the transcendental number

1.63252691943815284477....

This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either

is rational, which proves the theorem, or it is irrational (as it turns out to be), and then

is an irrational to an irrational power that is rational, which proves the theorem. The proof is not constructive, as it does not say which of the two cases is true, but it is much simpler than Kuzmin's proof.


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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#22 2021-11-06 03:54:53

ganesh
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Posts: 34,872

Re: Some special numbers

33) Cahen's constant


In mathematics, Cahen's constant is defined as an infinite series of unit fractions with alternating signs:

Here

denotes Sylvester's sequence, which is defined recursively by
and

Combining these fractions in pairs leads to an alternative expansion of Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence. This series for Cahen's constant forms its greedy Egyptian expansion:

This constant is named after Eugène Cahen (also known for the Cahen-Mellin integral), who was the first to introduce it and prove its irrationality.


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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#23 2021-11-06 17:10:51

ganesh
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Registered: 2005-06-28
Posts: 34,872

Re: Some special numbers

34) Fermat number

Pierre de Fermat (between 31 October and 6 December 1607 - 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica. He was also a lawyer at the Parlement of Toulouse, France.

No. of known terms  :  5
Conjectured no. of terms  :  5

Subsequence of    Fermat numbers
First terms  :  3, 5, 17, 257, 65537
Largest known term  :  65537

In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form

where n is a non-negative integer. The first few Fermat numbers are:

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ...
If

is prime and k > 0, then k must be a power of 2, so
is a Fermat number; such primes are called Fermat primes. As of 2021, the only known Fermat primes are
and
heuristics suggest that there are no more.

Basic properties

The Fermat numbers satisfy the following recurrence relations:



for n ≥ 1,


for n ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both

and Fj; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd.

Further properties

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers

are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed that


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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#24 2021-11-14 22:29:29

ganesh
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Registered: 2005-06-28
Posts: 34,872

Re: Some special numbers

35) Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in number theory. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.


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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#25 2021-11-23 00:41:46

ganesh
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Registered: 2005-06-28
Posts: 34,872

Re: Some special numbers

36) Kaprekar's constant

6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following rule:

* Take any four-digit number, using at least two different digits (leading zeros are allowed).
* Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
* Subtract the smaller number from the bigger number.
* Go back to step 2 and repeat.

The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 1495:

9541 – 1459 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174

The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4.

Other "Kaprekar's constants"

There can be analogous fixed points for digit lengths other than four, for instance if we use 3-digit numbers then most sequences (i.e., other than repdigits such as 111) will terminate in the value 495 in at most 6 iterations. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Kaprekar constants".

Other properties

* 6174 is a Harshad number, since it is divisible by the sum of its digits.
* 6174 is a 7-smooth number, i.e. none of its prime factors are greater than 7.
* 6174 can be written as the sum of the first three degrees of 18:

The sum of squares of the prime factors of 6174 is a square:


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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