 ## Play With Numbers – Part Three

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples . . .

John B. Conway

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 two previous blogposts Play With Numbers – Part One (link is available here https://math1089.in/2021/01/18/play-with-numbers-part-one/) and Play With Numbers – Part Two (link is available here https://math1089.in/2021/03/15/play-with-numbers-part-two/).

Polygonal numbers are number representing dots that are arranged into a geometric figure. Starting from a common point and augmenting outwards, the number of dots utilized increases in successive polygons. As the size of the figure increases, the number of dots used to construct it grows in a common pattern.

The most common types of polygonal numbers take the form of triangles and squares because of their basic geometry. Above figure illustrate examples of the first four polygonal numbers – the triangle, square, pentagon, and hexagon.

The Lucas numbers are the sequence of integers {Ln} defined by the (linear recurrence) relation Ln = Ln − 1 + Ln – 2 with the initial conditions L1 = 1 and L2 = 3.

The first few Lucas numbers are given by 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, . . .

If we consider the ratios of successive Lucas numbers, and no matter what two positive values we start with, we will eventually end up with ф (phi) = 1·6180339…

A Lucas prime is a Lucas number that is prime. The first few Lucas primes are 2, 3, 7, 11, 29, 47, 199, 521, . . .

The Lucas spiral, made with quarter arcs, is a good approximation of the golden spiral (which is a logarithmic spiral whose growth factor is ф, the golden ratio) when its terms are large. However, when the terms become very small, the arc’s radius decreases rapidly from 3 to 1 then increases from 1 to 2.

Repunit is an integer written (in decimal notation) as a string of 1’s. Few examples are 11, 111, 1111 etc.

We normally use the symbol Rn to denote the repunit consisting of n consecutive 1’s. Normally for a n digit repunit, we have

First few repunits are given below:

• R1 = 1
• R2 = 11
• R3 = 111 = 3 × 37
• R4 = 1111 = 11 × 101
• R5 = 11111 = 41 × 271
• R6 = 111111 = 3 × 7 × 11 × 13 × 37
• R7 = 1111111 = 239 × 4649
• R8 = 11111111 = 11 × 73 × 101 × 137

For a repunit Rn to be prime, the subscript n must be a prime; that this is not a sufficient condition is shown by R5, R7.

In fact, if n | m then Rn | Rm

Demlo Numbers are the numbers 1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 1234567900987654321, 123456790120987654321, 12345679012320987654321, 1234567901234320987654321, . . .

First nine of these are the square of the repunits 1, 11, 111, 1111, . . . , 111111111. Indian mathematician Kaprekar studied the Demlo numbers, named after a train station thirty miles from Bombay on the then G. I. P. Railway where he had the idea of studying them.

Moreover, the sum of the digits for the first few Demlo numbers are 1 = 12, 4 = 22, 9 = 32, 16 = 42, 25 = 52, 36 = 62, 49 = 72, 64 = 82, 81 = 92, 82, 85, 90, . . .

Below we can see how first nine Demlo numbers be arranged:

Hexagonal number represents the number of dots that can be arranged evenly in a hexagon.

A hexagonal number is a figurate number. The general formula for the nth hexagonal number is Hn = n(2n − 1). The first few hexagonal numbers are 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, . . .

Every hexagonal number is a triangular number, but every triangular number is not a hexagonal number.

Transcendental number is a (possibly complex) number that is not the root of any integer polynomial. In other words, it is not an algebraic number of any degree.

The best known examples are π and e.

Every real transcendental number must be irrational, but every irrational number is not transcendental.

For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0.

The golden ratio (denoted by ф) is another irrational number that is not transcendental, as it is a root of the polynomial equation x2x − 1 = 0.

Happy number is a number which eventually reaches 1 (we can say that 1 is a fixed point) when replaced by the sum of the square of each digit.

For example, 13 is a happy number because 12 + 32 = 10 and 12 + 02 = 1.

The first few happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, . . .  In this sequence, we can see that 1, 10, 100 etc. are happy numbers. Likewise, 7 and 70. The (reverse) numbers like 13, 31; 19, 91; 23, 32; 28, 82; 49, 94; 68, 86; 79, 97 are also happy numbers.

A number which is not happy is called unhappy or sad.

For example, 11 is a unhappy (sad) number because we never reach to 1. See the following sequence:

• 12 + 12 = 2
• 22 = 4
• 42 = 16
• 12 + 62 = 37
• 32 + 72 = 58
• 52 + 82 = 89
• 82 + 92 = 145
• 12 + 42 + 52 = 42
• 42 + 22 = 20
• 22 + 02 = 4
• 42 = 16

Dudeney Number is a positive integer for which the sum of its decimal digits is equal to the cube root of the number.

For example, 512 is a Dudeney number because it is the cube of 8, which is the sum of its decimal digits 5 + 1 + 2 = 8.

19683 is another example of Dudeney number because it is the cube of 27, which is the sum of its decimal digits 1 + 9 + 6 + 8 + 3 = 27.

Kaprekar’s Number is the number 6174 (details can be found at (https://math1089.in/2021/02/01/lets-play-with-6174-the-four-digit-kaprekars-constant/) and at (https://math1089.in/2021/01/23/converging-to-495-the-three-digit-kaprekars-constant/). The Indian mathematician D. R. Kaprekar made the following discovery in 1949.

(1) Take a four-digit number with different digits (acbd with .a < b < c < d)..
(2) Form the largest and the smallest number from these four digits (they are dcba and abcd)..
(3) Find the difference of these digits. May be this is 6174 (dcbaabcd = 6174).
If it is not, form the largest and the smallest  number from the difference and subtract these numbers again. We may have to repeat this procedure.
The end result is always 6174, but there are no more than 7 steps.

Sum-product Number is a natural number that is equal to the product of the sum of its digits and the product of its digits.

For example, the number 144 is a sum-product number, because 1 + 4 + 4 = 9, (1)(4)(4)=16 and  (9)(16)=144.

Obviously, such a number must be divisible by its digits as well as the sum of its digits. There are only three sum-product numbers, namely 1, 135 = (1 + 3 + 5)(1 ×number, 3 × 5) and 144.

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