Definitions - Notations

Every Number Should Be Significant & Meaningful!

A page of the Numeropedia - the Special Encyclopedia of Numbers

Numbers in Intervals  & All Numbers (1)All Numbers (2)  & All Numbers (3)

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DEFINITIONS of Mathematical Terms

- Algebraic numberis 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 2n that contains the digits 666: e.g., 2157, 2192, 2218, 2220, 2222, 2224, 2226, 2243, 2245, 2247, 2251,…, 2666,…

- An arithmetic progression with common difference c is a sequence of numbers in the the form: a0+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 3rd powers (and eventually all odd powers) ending with the number itself.

- Bernoulli numbers are defined by the infinite series of the function: x/(ex–1) = 1+B1(x/1!)+B2(x2/2!)+B3(x3/3!)+… They are related to the Riemann zeta function z(s)over complex variable s by the formula: Bn = (–1)n+1n´z(1–n).

- Brown number is a square number, which is a factorial plus 1, e.g.: 25 = 52 = 4!+1, 121 = 112 = 5!+1 and 5041 = 712 = 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) = (n2+n+2)/2.

- Carmichael numberis an odd composite number satisfying Fermat’s little theorem: for any number a relatively prime to n, the number (an–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 2n–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: Cn = C(2n,n)/(n+1) = (2n)!/(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 i2 = –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´2n+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 Ei) are defined by the infinite series: sec(x) = 1/cos(x) = 1+E2(x2/2!)+E4(x4/4!)+E6(x6/6!)+…, where E0 = 1, E2 = –1, E4 = 5, E6 =  –61, E8 = 1,385, E10 = –50,521, E12 = 2,702,765, E14 = –199,360,981, E16 = 19,391,512,145, E18 = –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^(2n)). 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 n³ 5.

- Fibonacci numbers: A number in the Fibonacci sequence equals the sum of 2 previous numbers: F0 = 0, F1 = 1 and for n³ 2,Fn= Fn–1+Fn–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 i2 = –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 (21/2, 53/7…) 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: L1 = 1, L2 = 3 and for n³ 3,Ln= Ln–1+Ln–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, x2+y2+z2 = 3xyz. 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 2n–1.

Mersenne prime number is a prime number of the form 2p–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…lmn such that its multiple with n (1-digit) is nabc…lm (or even the alternating version bc…lmna).

. A p-pseudoparasite number of the form abc…lmn such that its multiple with p (1-digit) is nabc…lm, where p¹n (or even the alternating version bc…lmna).

. A p-pseudoparasite-double number of the form abc…lmn such that its multiple with p is mnabc…l (or even the alternating version c…lmnab).

- 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 2n–1(2n–1) is a perfect number if and only if (2n–1) is a prime number. It must be a triangular number: sum of all integers from 1 to (2n–1) and its last digit is either 6 or 8. The first perfect numbers are: 6, 28, 496, 8128, 212(213–1) = 33550336, 216(217–1) = 8,589,869,056 and 218(219–1) = 137,438,691,328… Also: amicable numbers and sociable number chain.

Also: Abundantnumber is a number smaller than to the sum of its aliquot divisors. Deficient 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 = 2592 = 25´92. 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 n2: P(n) = n(n+1)(2n+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×2n–1 is composite, for every integer n³ 1.

- Sierpinski number of first kind is a prime number of the form nn+1. There are only 3 such numbers: 2 = 11+1, 5 = 22+1 and 257 = 44+1.

. Sierpinski number of 2nd kind is a positive odd number k, such that the number k×2n+1 is composite, for every integer n³ 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 Germaineprime 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 = 246×320×59×76×112×133×17×19×23×29×31×41×47×59×71.

- Transcendental numberis 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 nth root number is 6 times the (n–1)th root minus the (n–2)th root: 1, 62 = 36, 352 = 1225, 2042 = 41616, 11892 = 1,413,721, 69302 = 48,024,900, 40,3912 = 1,631,432,881, 235,4162 = 55,420,693,056, 1,372,1052, 7,997,2142, 46,611,1792, 271,669,8602, 1,583,407,9812, 9,228,778,0262, 53,789,260,1752 and 313,506,783,0242

- 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 (2n) 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 (2p–1–1) is divisible by p2. 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 p2. The only known Wilson prime numbers (up to 5´108) are: 5, 13 and 563.

- Woodall prime numbersare prime number of the form n´2n–1. The first Woodall prime numbers are withn = 2, 3, 6, 30, 75, 81…

Notations

.  ëxû is the largest integer less than x (or the intergral part of x), also denoted by [x] or called floor function of x.

.  éxù is the smallest integer greater than x, also called the ceiling function of x.

.  a^n = an:the nth power of a, or the product (multiplication) of n terms of a.

.  Sn is the symbol representing an infinite sum of terms where n runs over all positive integers from 1 to infinity.

.Pnis the symbol representing an infinite product of terms where n runs over all positive integers from 1 to infinity.