The set of positive integers can be subdivided into several types or classes of numbers. Each type is a subset of the set of positive integers, however many of these sets overlap, by which I mean that they have integers in common. To follow is a list of as many of these sets as I could think of at this time. This is an introductory list that provides only a brief description of each number type. Of course there is no limit to the number of types, since there are an infinite number of combinations of the operators and integers. The ones listed are the more commonly known types. Just to make this page a bit more interesting, I will use the letter A for the digit ten, and express all numbers in base ten mostly using the Alternate Number System.
*** Prime and Composite numbers ***
These two sets combine to include all positive integers (except 1). Primes are those numbers that have only 1 and itself as factors. If a number is not prime, it is composite. Prime = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, ... Composite = 4, 6, 8, 9, A, 12, 14, 15, 16, 18, 1A, 21, 22, ... The number 1 is classified either as prime or composite, depending on who you talk with. Primes are far rarer than composites, however Euclid proved that there are an infinite number of primes.*** Even and Odd (Generally Modular) ***
Even = 2, 4, 6, 8, A, 12, 14, ... Odd = 1, 3, 5, 7, 9, 11, 13, ... More generally N == a(mod b) and 1 <= a <= b Odd numbers == 1(mod 2) Even == 2(mod 2) Notice that I replace 0(mod b) with b(mod b) to eliminate zero. Examples : 1(mod 3) = 1, 4, 7, A, 13, 16, 19, 22, ... 2(mod 3) = 2, 5, 8, 11, 14, 17, 1A, 23, ... 3(mod 3) = 3, 6, 9, 12, 15, 18, 21, 24, ...*** Fermat Numbers and Primes *** These are numbers of the form (2^n + 1) such that n is a power of 2. Fermat thought that all numbers of this form were prime, since n = 2, 4, 8 and 16 are prime, however, no other primes have ever been found!
*** Mersenne Numbers and Primes ***
Mersenne numbers = (2^n - 1) where n is the number of digits = 1 in base 2. Mersenne numbers are prime only if n is prime. For example, (2^7 - 1) = 1111111(base 2) = 127 is prime and n=7 is prime. But 2^n - 1 is not necessarily prime when n is prime Example : 2^11 - 1 = 1A47 = 23 x 89 even though n=11 is prime.*** Repunit Numbers and Primes ***
Repunits numbers = (10^n-1)/9 where n is the number of digits = 1. Repunits are prime only if n is prime. For example, 11, 1111111111111111111(base 10) are both prime. The longer number has 19 ones. But 10^n - 1 is not necessarily prime when n is prime. Example : (10^3-1)/9 = 111(base 10) = 37 x 3 even though n=3 is prime.*** General numbers of the Mersenne/Repunit type ***
G(b,n) = (b^n - 1)/(b - 1) I use the symbol G to represent these numbers since they expand to form a Geometric Progression = b^(n-1) + b^(n-2) + ... + b^3 + b^2 + b + 1. In base b number system, these numbers are 11....11 with n digits equal to 1. When b=2, we have Mersenne numbers and when b=10 we have repunits. Once again, these numbers are prime only if n is a prime. For example, (5^3 - 1)/4 = 111(base 5) is prime and n=3 is prime. Here are two primes with n = b: (19^19 - 1)/18 and (31^31 - 1)/30 But G(b,n) is not necessarily prime when n is prime Example : (3^5 - 1)/2 = 121 = 11 x 11 even though n=5 is prime.*** Fibonacci Numbers = F(n) ***
F(n) = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... These are determined using the recursive relation: F(n+2) = F(n+1) + F(n) where F(1) = F(2) = 1 There is a lot of interesting material that has been published on these numbers over the centuries. There are also many web sites that discuss some of the properties of Fibonacci Numbers. Generally : T(n+2) = T(n+1) + T(n), T(1)=a, T(2)=b Fibonacci numbers have T(1) = T(2) = 1 but are usually written using F not T Lucas numbers are also of this type, and have T(1)=1 and T(2)=3 Also Fibonacci primes = F(p) = prime, only if p = prime (except F(4)=3). F(p) = 2, 3, 5, 13, 89, 233, 1597, 28657, ...*** Powers = b^n ***
2^n = 2, 4, 8, 16, 32, 64, 128, 256, 512, A24, 1A48, ... 3^n = 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ... 5^n = 5, 25, 625, 3125, 15625, 78125, 38A625, ...*** Pythagorean Triples ***
These are triplets of integers (a,b,c) that satisfy the equation: a^2 + b^2 = c^2 Usually these are considered such that a,b and c are relatively prime, since the others reduce to this form. There are an infinite number of such triplets. The most common one is (3,4,5) ==> 3^2 + 4^2 = 5^2. Other examples are (5,12,13) and (20,21,29) and so on. Another related equation that I believe is called Hoppenot's Rule, says that the sum of squares of (n+1) consecutive integers, of which 2n(n+1) is the greatest equals the sum of squares of the next n integers. For example (n=2) : 10^2 + 11^2 + 12^2 = 13^2 + 14^2 Another is (n=3) : 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2*** Factorials = n! ***
n! = product of successive integers = 1 x 2 x 3 x 4 x 5 x ... x n n! = 1, 2, 6, 24, 11A, 71A, 4A3A, 4A31A, 36287A ... they get big fast!*** Prime Product Factorials = p!! ***
p!! = product of successive primes = 2 x 3 x 5 x 7 x 11 x ... x p p!! = 2, 6, 2A, 1AA, 22AA, 29A2A, 4AA4AA, ...*** Binomial Numbers ***
B(n,1) = 1, B(n,n) = 1 B(n,k) = B(n-1,k-1) + B(n-1,k) Binomial numbers are the coefficients in the expansion of (a + b)^(n-1) The coefficients create the so-called Pascal triangle shown below, however there are records in Chinese documents of this triangle long before Pascal was born! 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 A A 5 1 For example : B(6,3) = B(5,2) + B(5,3) = 4 + 6 = A*** Stirling Numbers of the Second Kind ***
S(n,1) = 1, S(n,n) = 1, S(n,k) = S(n-1,k-1) + kS(n-1,k) (2 <= k <= n-1) These are easier to see in a Pascal-type triangle as follows: 1 1 1 1 3 1 1 7 6 1 1 15 25 A 1 1 31 8A 65 15 1 1 63 2A1 34A 13A 21 1 For example : S(7,4) = S(6,3) + 4S(6,4) = 8A + (4 x 65) = 34A (Stirling numbers of the first kind include negative integers, therefore I have omitted them)*** Perfect Numbers *** Perfect numbers are those whose factors add up to the number. Perfect = 2^(p-1)(2^p - 1) = 6, 28, 496, 8128, ... For example, p = 3 produces : 2^(3-1)(2^3 - 1) = 4 x 7 = 28 28 has factors = 1, 2, 4, 7, 14 and these add up to 28. Perfect numbers are related to the previously mentioned Mersenne primes, since the number is perfect if and only if (2^p - 1) is prime. Numbers whose factor sum is less than the number are called deficient Numbers whose factor sum is greater than the number are called abundant
*** Amicable numbers ***
These numbers are called amicable because the sum of the factors of one equals the other number. The smallest of which is (21A, 284). Another pair is (1184, 11AA). Some think that if you provide someone with one of these numbers, and you retain the other, then your love is guaranteed - hence the name.*** Palindromes ***
Palindromes are numbers that are symmetrical about the middle. Examples are : 1221, 523325, 745212547, and so on.*** Figurate numbers ***
These numbers have geometric names because of the patterns created when the numbers creating them are arranged as dots. For example, the triangular pattern is: x xx xxx xxxx xxxxx Triangular = 1, 3, 6, A, 15, ... = sum of consecutive integers = n(n+1)/2 = 1 + 2 + 3 + 4 + ... Gnomons = 1, 4, 9, 16, 25, ... = sum of odd numbers = n^2 = 1 + 3 + 5 + 7 + ... Oblong = 2, 6, 12, 1A, 2A, ... = sum of even numbers = n(n+1) = 2 + 4 + 6 + 8 + ... Pentagonal = 1, 5, 12, 22, 35, ... = sum of 1(mod 3) numbers = n(3n-1)/2 = 1 + 4 + 7 + 10 + ... Hexagonal = 1, 6, 15, 28, 45, ... = sum of 1(mod 4) numbers = n(2n-1) = 1 + 5 + 9 + 13 + ... and so on ...*** Kaprekar's constants ***
These are created by arranging the digits of an n-digit number in ascending and descending order, subtracting the two numbers, and repeating this process until a constant is obtained. The 3-digit Kaprekar constant is 495 (ie. 954 - 459 = 495) The 4-digit Kaprekar constant is 6174 (ie. 7641 - 1467 = 6174) It's interesting to note that using the Alternate Number System, 7A82 also occurs for 4-digit numbers. (ie. A872 - 278A = 7A82)*** Carmichael Numbers ***
Fermat's Lesser Theorem says that (b^(n-1) - 1) has n as a factor if n is prime. However some non-prime values of n also work. Carmichael Numbers are special composites of this form. Carmichael Numbers are numbers (n) that were first discovered in 1909 by R.D. Carmichael. They are numbers that are factors of (b^(n-1) - 1) for every value of b as long as b and n are relatively prime. The first such number is n = 561 = 3 x 11 x 17 Two other examples are : n = AA5 = 5 x 13 x 17 and n = 1729 = 7 x 13 x 19 These numbers are quite rare, however several properties of them have been found. For example, all Carmichael numbers are odd, and have at least three different prime factors.*** Partition Numbers ***
Any integer (N) can be "split" or partitioned into from 1 to N parts. For example, 5 can be split into 7 such partitions: (1,1,1,1,1), (1,1,1,2), (1,1,3), (1,2,2), (1,4), (2,3), and 5 itself. The symbol p(N) is used to represent the total number of ways that N can be partitioned. For successive values of N, we have the following sequence: P(N) = 1, 2, 3, 5, 7, 11, 15, 22, 2A, 42, 56, 77, A1, 135, ... There is an amazing recursive formula to provide the nth term: Let p(0)=1 Stop the following process when the index of a term becomes less than zero. p(n) = p(n-1) + p(n-2) (m = 1) -p(n-5) - p(n-7) (m = 2) +p(n-12) + p(n-15) (m = 3) - ................ +- p(n-m(3m-1)/2) +- p(n-m(3m+1)/2) (m >= 1) The last line is the general form for the pairs of terms. When m is odd, add the two terms, and when m is even, subtract them. For example: p(12)= p(11) + p(10) - p(7) - p(5) + p(0) = 56 + 42 - 15 - 7 + 1 = 77
Enjoy.