*Every Number Should Be Significant & Meaningful!*

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**DEFINITIONS of Mathematical Terms**

- **Algebraic **number*is a root of some polynomial equation with integer coefficients. An algebraic number can be either irrational number or rational number. *

- **Amicable **number: The sum of all proper divisors of an amicable number including 1 (or also called aliquot divisors) is equal to its mate number and vice versa. Also, **perfect numbers** and **sociable number chain. **(It is unknown whether there exists any pair of amicable numbers, in which one is odd and one is even).

**. Amicable triplet**: sum of the aliquot divisors of any number in the triplet is the sum of the other two numbers.

- **Apocalypse **numberis a number having 666 digits.

- **Apocalyptic **numberis a number of the form 2* ^{n}* that contains the digits 666: e.g., 2

^{157}, 2

^{192}, 2

^{218}, 2

^{220}, 2

^{222}, 2

^{224}, 2

^{226}, 2

^{243}, 2

^{245}, 2

^{247}, 2

^{251},…, 2

^{666},…

- An **arithmetic progression** with common difference *c* is a sequence of numbers in the the form: *a*_{0}+*c*´*n*, for *n* = 0, 1, 2… That is the difference between two consecutive terms is always equal to the constant *c*.

- **Automorphic **number has its powers ending with the number itself. There are two 17-digit automorphic numbers: 43,740,081,787,109,376 and 56,259,918,212,890,625. They remain automorphic when dropping the left-most digits. Remark: **Trimorphic number** has its 3^{rd} powers (and eventually all odd powers) ending with the number itself.

- **Bernoulli **numbers are defined by the infinite series of the function: *x*/(*e ^{x}*–1) = 1+

*B*

_{1}(

*x*/1!)+

*B*

_{2}(

*x*

^{2}/2!)+

*B*

_{3}(

*x*

^{3}/3!)+… They are related to the Riemann zeta function z(

*s*)over complex variable

*s*by the formula:

*B*= (–1)

_{n}

^{n}^{+1}

*n*´z(1–

*n*).

- **Brown number** is a square number, which is a factorial plus 1, e.g.: 25 = 5^{2} = 4!+1, 121 = 11^{2} = 5!+1 and 5041 = 71^{2} = 7!+1. Mathematician Paul Erdös conjectured that these known numbers are the only 3 Brown numbers.

- **Cake number** (or **pizza number**) is the maximum number of pieces in which a (circular and flat) cake can be cut by a straight knife *n* times, *C*(*n*) = (*n*^{2}+*n*+2)/2.

- **Carmichael **numberis an odd composite number satisfying Fermat’s little theorem: for any number *a *relatively prime to *n*, the number (*a ^{n}*

^{–1}–1) is divisible by

*n*. It is called an absolute pseudoprime number (pseudoprime to any base).

**. **Remark: An ancient Chinese conjecture: “*n* is a prime number if and only if *n* divides the number 2* ^{n}*–2” was wrong, by counterexamples: number 341 and 561…

- **Catalan** number is the number of ways to cut an (*n*+2)-side polygon into *n* non-intersecting triangles: *C _{n}* =

*C*(2

*n*,

*n*)/(

*n*+1) = (2

*n*)!/(

*n*!(

*n*+1)!), where the combinatoric

*C*(

*m*,

*n*) is the number of ways to form a group of

*n*items among

*m*items.

- **Complex **number is the addition of two components, a real number and a pure imaginary number, in the form (*a*+*bi*), where *a* and *b* are real numbers and *i*^{2} = –1 or *i* = (–1)^{1/2}. Then: the real part Â(*a*+*bi*) = *a* and the imaginary part Á(*a*+*bi*) = *b*.

- **Composite **number is a number that can be factored into 2 or more prime numbers, i.e. a product of 2 or more prime numbers.

- **Cullen prime **numbers are prime numbers of the form *n*´2* ^{n}*+1. The only known Cullen prime numbers are with

*n*= 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275 and 481899. Masakatu Morii found the last number on 30 September 1998.

- **Dice **number is a number formed by a 4-face rotation of a 6-face dice.

- “**Emirp”** (“prime” spelled backward) is a prime number (which is not a palindromic prime) such that its digit-researsal number is also a prime number.

- **Euler **numbers (or **secant **numbers *E _{i}*)

**are defined by the infinite series: sec(**

*x*) = 1/cos(

*x*) = 1+

*E*

_{2}(

*x*

^{2}/2!)+

*E*

_{4}(

*x*

^{4}/4!)+

*E*

_{6}(

*x*

^{6}/6!)+…, where

*E*

_{0}

^{ }= 1,

*E*

_{2}= –1,

*E*

_{4}= 5,

*E*

_{6}= –61,

*E*

_{8}= 1,385,

*E*

_{10}= –50,521,

*E*

_{12}= 2,702,765,

*E*

_{14}= –199,360,981,

*E*

_{16}= 19,391,512,145,

*E*

_{18}= –2,404,879,675,441… The odd-indexed Euler numbers are all 0. The even-indexed Euler numbers have alternating signs.

- **Factorial of n**(

*n*factorial) is defined by the product of

*n*consecutive numbers from 1 to

*n*:

*n*! = 1×2×…×

*n*.

- **Fermat **number is a number of the form (1+2^(2* ^{n}*)). The first 5 Fermat numbers 3, 5, 17, 257 and 65,537 (for

*n*= 0 to 4) are all prime numbers. Only composite Fermat numbers are known for

*n*³ 5.

- **Fibonacci **numbers: A number in the Fibonacci sequence equals the sum of 2 previous numbers: F_{0} = 0, F_{1} = 1 and for *n*³ 2,F_{n}_{}= F_{n}_{–1}+F_{n}_{–2}. The first Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34…

- **Harshad **number(or **Niven **number) is a number divisible by sum of its digits.

- **Happy number** is a number whose sum of the squares of the digits eventually equals 1.

- **Honest number** is a number *n* that can be described using exactly *n* letters in standard mathematical English. Conjecture: All numbers greater or equal to 13 is honest.

- **Imaginary **number (or **pure imaginary **number) is a complex number with no real component, i.e. of the form *bi*, where *b* is a real number and *i*^{2} = –1 or *i* = (–1)^{1/2}.

- **Integer** is a whole number, positive or negative and 0.

- **Irrational **number is not a root of any monomial (linear polynomial) equation with integer coefficients. An irrational number can be either algebraic (2^{1/2}, 5^{3/7}…) or transcendental (*e*, *pi* π…).

- **Keith** number is an *n-*digit integer *N *such that if a Fibonacci-like sequence (each term of it is the sum of the *n* previous terms) is formed which the first *n* terms taken as the digits of the number *N*, then the number *N* will occur as a term in that sequence.

- **Lucas **number: A number in the Lucas sequence equals the sum of 2 previous numbers: L_{1} = 1, L_{2} = 3 and for *n*³ 3,L_{n}_{}= L_{n}_{–1}+L_{n}_{–2}. The first Lucas numbers are: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521…

- **Lychrel number **is a number that does not produce a palindromic number by applying the **196**-algorithm (or **reversal-addition** algorithm): repeated adding the number itself with its digit-reversal number).

- **Markov number** (Markoff number) is a positive integer *x*, *y* or *z* that is part of a solution to the Markov Diophantine equation, *x*^{2}+*y*^{2}+*z*^{2} = 3*xyz*. The first few Markov numbers are: 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325…

- **Mersenne **number is a number of the form 2* ^{n}*–1.

**Mersenne prime **number is a prime number of the form 2* ^{p}*–1, obviously the exponent

*p*must be a prime number.

- **Natural **number is a positive whole number or positive integer.

- **Palindromic** number is a number that is the same when written backwards. It is conjectured that 196 is the smallest number that will never form a palindromic number by the **196**-algorithm (or **reversal-addition** algorithm): repeated adding the number itself with its digit-reversal number).

- **Pan-digital **number is a number with all 10 digits 0-9 appearing once and the first digit is not 0. **Zero-less pan-digital **number is a number with all 9 digits 1-9 appearing once and the first digit is not 0.

- **Pan (Depression) prime **number is a prime number having all same interior digits, which are smaller than its two end-digits. Also, Plateau prime number.

- **Parasite/Pseudoparasite **numbers.

**. **An **n-****parasite **number of the form *abc…lm n* such that its multiple with

*n*(1-digit) is

*(or even the alternating version*

__n__abc…lm*bc…lmna*).

**. **A **p-****pseudoparasite** number of the form *abc…lm n* such that its multiple with

*p*(1-digit) is

*, where*

__n__abc…lm*p*¹

*n*(or even the alternating version

*bc…lmna*).

**. **A **p-****pseudoparasite-double** number of the form *abc…l mn* such that its multiple with

*p*is

*(or even the alternating version*

__mn__abc…l*c…l*).

__mn__ab- **Perfect **number is a number equal to the sum of all of its proper divisors, including 1 (or also called aliquot divisors). Every even number of the form 2^{n}^{–1}(2* ^{n}*–1) is a perfect number if and only if (2

*–1) is a prime number. It must be a triangular number: sum of all integers from 1 to (2*

^{n}*–1) and its last digit is either 6 or 8. The first perfect numbers are: 6, 28, 496, 8128, 2*

^{n}^{12}(2

^{13}–1) = 33550336, 2

^{16}(2

^{17}–1) = 8,589,869,056 and 2

^{18}(2

^{19}–1) = 137,438,691,328… Also:

**amicable numbers**and

**sociable number chain.**

** **Also: **Abundant**number is a number *smaller* than to the sum of its aliquot divisors. **D****eficient** number is a number *bigger* than to the sum of its aliquot divisors.

- **Polite** number is a positive number that can be written as the sum of 2 or more consecutive number. E.g.: 7 = 3+4 is a polite number while 8 is an impolite number.

- **Prime **number is a number that can be divisible only by 1 and itself.

**. Absolute prime** number (**permutable prime **number)is a prime number, which remains a prime number after permuting its digits.

** . Circular prime **number, a special case of absolute prime number, is a prime number, which remains a prime number when circulating its digits.

**. Plateau prime **number is a prime number having all same interior digits, which are larger than its two end-digits. Also, Pan (or Depression) prime number.

- **Printer’s error number **does not change its value when any of its digits are replaced by an exponent of the same value of those digits at the same positions. The first known case is: 2592 = 2^{5}9^{2} = 2^{5}´9^{2}. It is still true for 0’s added to the right side of the number.

- **Pyramidal **number is a sum of squares of all integers from 1 to *n*^{2}: *P*(*n*) = *n*(*n*+1)(2*n*+1)/6.

- **Rare number **is a number that gives a perfect square number by adding as well as subtracting its reverse.

- **Rational **number is a root of some monomial (linear polynomial, of degree 1 only) equation with integer coefficients, i.e., the number can be written as a ratio of two integers. E.g.: 2/5, 7/11… A rational number is obviously an algebraic number.

- **Real **number is defined to be associated with a point on a line, either rational number or irrational number.

- **Riesel **number is a positive odd number *k*, such that the number *k*×2* ^{n}*–1 is composite, for every integer

*n*³ 1.

- **Sierpinski number of first kind** is a prime number of the form *n ^{n}*+1. There are only 3 such numbers: 2 = 1

^{1}+1, 5 = 2

^{2}+1 and 257 = 4

^{4}+1.

**. Sierpinski number of 2 ^{nd} kind** is a positive odd number

*k*, such that the number

*k*×2

*+1 is composite, for every integer*

^{n}*n*³ 1.

- **Smith **numberis a composite number whose sum of its digits is equal to the sum of digits of all of its prime factors.

- **Sociable **number chain: The sum of all proper divisors of a sociable number including 1 (or also called aliquot divisors) is equal to the next number in the chain. Hence a pair of **amicable** numbers is a 2-link chain, while a **perfect** number is 1-link chain. No 3-link chain has been found yet.

- **Sophie Germaine****prime** number is an odd prime number such that twice of it plus 1 is also a prime number.

- **Sphenic number **is a number that has precisely 3 distinct prime factors. The first ten sphenic numbers are: 30, 42, 66, 70, 78, 102, 105, 110, 114 and 130.

- **Subfactorial** of *n *(!*n* = *n*!×[1–(1/1!)+(1/2!)–…(1/*n*!)]) is the number of permutations of *n* objects in which no object appears in its natural position, i.e. the number of “derangement”. By recurrence relations: !*n* = *n*×(*n*–1)+(–1)* ^{n}* or !(

*n*+1) =

*n*×[!

*n*+!(

*n*–1)].

- **Supersingular prime numbers** are the 15 prime factors of the order of the Monster group *M*: 808017424794512875886459904961710757005754368000000000 = 2^{46}×3^{20}×5^{9}×7^{6}×11^{2}×13^{3}×17×19×23×29×31×41×47×59×71.

- **Transcendental **number*is an irrational number, which is not a root of any polynomial equation with integer coefficients. It must be not an algebraic number. E.g.: Euler number **e*, *pi *π…

- **Triangular **number is a sum of all integers from 1 to *n*: T(*n*) = *n*(*n*+1)/2. Generally, there are **polygonal** numbers, such as square number, pentagonal, hexagonal, heptagonal, octagonal… numbers.

**. **There are infinitely many square numbers that are also triangular numbers. The *n*^{th} root number is 6 times the (*n*–1)^{th} root minus the (*n*–2)^{th} root: 1, 6^{2 }= 36, 35^{2} = 1225, 204^{2 }= 41616, 1189^{2} = 1,413,721, 6930^{2} = 48,024,900, 40,391^{2} = 1,631,432,881, 235,416^{2} = 55,420,693,056, 1,372,105^{2}, 7,997,214^{2}, 46,611,179^{2}, 271,669,860^{2}, 1,583,407,981^{2}, 9,228,778,026^{2}, 53,789,260,175^{2} and 313,506,783,024^{2}…

- **Truncatable prime number**. If a zero-free prime number is still a prime number by successively removing the leftmost (rightmost or both) digits one by one, then it is called a left- (right- or bi-) truncatable prime number. The first 4 prime numbers (2, 3, 5 and 7) are bi-truncatable.

- **Twin prime** numbers are two prime numbers whose difference is 2, except the only twin prime (2, 3).

**. Cousin prime** numbers are two prime numbers whose difference is 4.

**. Sexy prime** numbers are two prime numbers whose difference is 6.

-** Unique prime **number is a prime number, other than 2 or 5, that there is no other prime number whose reciprocal has period of the same length.

- **Untouchable **number(defined by Paul Erdös) is a number that is never the sum of the proper divisors of any other number. A proper divisor of a number *N* is a number that divides *N*, (also called as a factor of *N*), excluding the number *N* itself. There are infinitely many untouchable numbers, proven by Paul Erdös. The first numbers are: 2, 5, 52, 88, 96, 120…

- **Vampire **number is of an even number of digits (2*n*) formed by multiplying a pair of *n*-digit numbers (called its fangs), whose digits are taken from the original number in any order. Pairs of trailing zeroes are not allowed. Obviously, there is no 2-digit vampire number since *a***´***b* is always less than *a***´**10, for any 2 digits *a* and *b*.

**. **Lemma: If the number *x*´*y *is a vampire number, than *x*´*y* = *x*+*y*, modulo 9.

**. Prime vampire **number is a vampire number whose fangs are prime numbers.

- **Wieferich prime **number is a prime numbers *p* such that (2^{p}^{–1}–1) is divisible by *p*^{2}. The only known Wieferich prime numbers are 1093 and 3511.

- **Wilson prime **numberis a prime number *p* such that the number (*p*–1)!+1 is divisible by *p*^{2}. The only known Wilson prime numbers (up to 5´10^{8}) are: 5, 13 and 563.

- **Woodall prime **numbersare prime number of the form *n*´2* ^{n}*–1. The first Woodall prime numbers are with

*n*= 2, 3, 6, 30, 75, 81…

**Notations**

**.** **ë x**

**û**is the largest integer less than

*x*(or the intergral part of

*x*), also denoted by [

*x*] or called floor function of

*x*.

**.** **é x**

**ù**is the smallest integer greater than

*x*, also called the ceiling function of

*x*.

**.** **a****^ n = a^{n}**:the

*n*power of

^{th}*a*, or the product (multiplication) of

*n*terms of

*a*.

**.** **S*** _{n}* is the symbol representing an infinite sum of terms where

*n*runs over all positive integers from 1 to infinity.

**.**P* _{n}*is the symbol representing an infinite product of terms where

*n*runs over all positive integers from 1 to infinity.