Play With Numbers – Part Two

One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics.

Leon Henkin

Glad you came by. I wanted to let you know I appreciate your spending time here at the blog very much. I do appreciate your taking time out of your busy schedule to check out Math1089!

This is the continuation to the previous blogpost Play With Numbers – Part One (link is available here https://math1089.in/2021/01/18/play-with-numbers-part-one/). One more blogpost will appear very soon with the title Play With Numbers – Part Three.

A figurate number is a number that can be represented by a regular geometrical arrangement of equally spaced points.

If the arrangement forms a regular polygon, the number is called a polygonal number.

In the above figure, the polygonal numbers respectively called triangular, square, pentagonal and hexagonal numbers.

A cubic number is a figurate number of the form n3 with n a positive integer.

In other words, if we multiply a number by itself and then by itself again, the result is a cube number.

First Five Cubic Numbers

They are named cubic (or cubed) numbers because they can also be used to calculate the volume of a cube. Since a cube has sides of the same length, width and height, we can calculate its volume by cubing the side length.

The first few are 1, 8, 27, 64, 125, 216, 343, . . . A list is given below for quick reference.

Cubes of first 20 numbers

A Krishnamurthy (or Peterson or Strong) number is a number whose sum of the factorial of digits is equal to the number itself.

For example, consider the number 145. Digits of this number are 1, 4, 5 and sum of factorial of the digits = 1! + 4! + 5! = 1 + 24 + 120 = 145. Hence, 145 is a Krishnamurthy number.

Few other examples of Krishnamurthy number are 1, 2, 40585.

The Fibonacci numbers (sometimes called pine cone numbers) are the sequence of numbers {Fn} defined by the (linear recurrence) relation Fn = Fn − 1 + Fn − 2 with the initial conditions F1 = F2 = 1.

The first few Fibonacci numbers are given by 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .

Fibonacci Numbers – Geometric Interpretation

The Fibonacci numbers Fn are squareful for n = 6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, . . . and squarefree for n = 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, . . .

The ratios of successive Fibonacci numbers Fn/ Fn − 1 approaches the golden ratio ф (phi) as n approaches infinity. The golden ratio is a root of the polynomial x2x – 1.

A number of the form Mn = 2n – 1 are known as Mersenne number.

The first few Mersenne numbers are 1, 3, 7, 15, 31, 63, 127, 255, . . .

A Mersenne prime is a prime number of the form 2n – 1.

Largest Mersenne Prime

The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, . . .

Pentagonal numbers represent the number of dots that can be arranged evenly in a pentagon.

First few pentagonal numbers are 1, 5 = 1 + 4, 12 = 1 + 4 + 7, 22 = 1 + 4 + 7 + 10, 35 = 1 + 4 + 7 + 10 + 13, . . . 

Pentagonal Numbers

A simple formula to derive the pentagonal numbers is pn = n(3n − 1)/2 for all n ≥ 1.

If pn denotes the nth pentagonal number, then pn = pn − 1 + (3n − 2) for all n ≥ 2, where p1 = 1.

1729 is commonly known as the Hardy Ramanujan number. It is the smallest number representable in two ways as a sum of two cubes:

  • 1729 = 1728 + 1 = 123 + 13
  • 1729 = 1000 + 729 = 103 + 93
1729 – Hardy Ramanujan Number

Once Prof. Hardy came to visit Ramanujan in a taxi whose number was 1729. There Ramanujan pointed out that 1729 is the smallest number that can be expressed as a sum of two cubes in two different ways.

There are an infinitely many such numbers. Few are

  • 4104 = 23 + 163 = 93 + 153
  • 13832 = 183 + 203 = 23 + 243

Amicable (or friendly) numbers are two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number.

The smallest pair of amicable numbers is (220, 284). Now proper divisors (a proper divisor of a number is a positive factor of that number other than the number itself) of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Sum of them = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. Proper divisors of 284 are 1, 2, 4, 71 and 142 and their sum = 1 + 2 + 4 + 71 + 142 = 220.  

Amicable Numbers – An Example

Few amicable pairs are (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), . . .

A number is termed as a tetrahedral (or triangular pyramidal) number if it can be represented as a pyramid with a triangular base and three sides, called a tetrahedron.

Tetrahedral numbers

The first ten tetrahedral numbers are 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, . . . The nth tetrahedral number is the sum of the first n triangular numbers.

An algebraic number is any real (or complex) number that is a solution of some single variable polynomial equation whose coefficients are all integers.

All rational numbers are algebraic. Examples include 35, 5/9, and −0.3445245245. Some irrational numbers are also algebraic. Examples are √3 and 41/3 (the cube root of 4). There are irrational numbers x for which no single variable, integer coefficient polynomial equation exists with x as a solution. Examples are π (the ratio of a circle’s circumference to its diameter in a plane) and e (the natural logarithm base). Numbers of this type are known as transcendental numbers. The set of algebraic numbers is denumerable.

While this is an abstract notion, theoretical mathematics has potentially far reaching applications in communications and computer science, especially in data encryption and security.

Star numbers are the number of cells in a generalized Chinese checkers board (or centered hexagram).

The nth star number is given by the formula Sn = 6n(n − 1) + 1. The first few star numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, . . .

Star Numbers

The star numbers satisfy the linear recurrence equation Sn = Sn − 1 + 12(n − 1).

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call Math1089 – Mathematics for All!“.

5 comments

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