Play with Numbers – Part One

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
Paul Halmos

Welcome to the blog Math1089 – Mathematics for All!.

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!

Square Numbers (also called perfect squares) are integers which is the square of an integer. In other words, it is the product of some integer with itself.

For example, 16 is a square number since it equals 4² and can be written as 4 × 4. 121 is another square number since it equals 11² and can be written as 11 × 11.

If we consider a square of side length n, then area of the square is n². Hence the number n is a square if and only if we can arrange n points in a square. Few examples are given below.

A square number can end only with the digits 0, 1, 4, 5, 6 or 9. Square of even numbers are even and square of odd numbers are always odd.

Perfect number, a positive integer that is equal to the sum of its proper divisors, excluding the number itself.

The smallest perfect number is 6 because 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6.

Similarly, 28 is a perfect number since 28 = 1 + 2 + 4 + 7 + 14. 496 is another perfect number since 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.

Triangular Number counts objects arranged in an equilateral triangle.  

The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers is 1, 3, 6, 10, 15, 21, 28, 36, 45, . . . , and a few of them are shown below (through diagrams).

Palindromic Number is a number that remains same when its digits are reversed.

General form of a palindromic number is a₁a₂a₃…a_n-2a_n-1a_n and every such number has reflectional symmetry across a vertical axis.

One example is 121. Few more examples are 77377, 12321, 555555, 123404321. Few palindromic primes are 101, 131, 151 etc. and few palindromic squares are 484, 10201, 12321 etc.

Smith Number is a composite number, the sum of whose digits is equal to the sum of the digits of its prime factors.

4 = 2 × 2 is the smallest Smith number. The sum of the digits of 4 is 4, and the sum of digits of its prime factors is 2 + 2 = 4. Another example is 666 = 2 x 3 x 3 x 37. Now 6 + 6 + 6 = 18 and 2 + 3 + 3 + (3 + 7) = 15, proving that 666 is a Smith number.

Few more examples are 22, 27, 58, 85, 94, 121, 166, 202, 265, 6036, 9942, 9975.

Automorphic Number (sometimes referred to as a circular number) is a natural number, the square of which ends in the same digits as the number itself. In other words, the number appears at the end of its square.

The numbers 5, 25, 76, 376 are automorphic because

• 5² = 2(5)
• 25² = 6(25)
• 76² = 57(76)
• 376² = 141(376)

Few more examples of automorphic numbers are 0, 1, 6, 625, 9376, 90625, 109376, 890625, 2890625.

Trimorphic Number is a natural number the cube of which ends in the same digits as the number itself. In other words, the number appears at the end of its cube.

Note that, all automorphic numbers are trimorphic. The numbers 4, 25, 51, 249 are trimorphic because

4³ = 6(4)

25³ = 156(25)

51³ = 1326(51)

249³ = 15438(249)

Few more trimorphic numbers are 0, 1, 5, 6, 9, 24, 49, 75, 76, 99, 125, 251, 375, 376.

Narcissistic Number (also known as an Armstrong number) is an n-digit number that is the sum of the nth powers of its digits.

It is evident from the definition that, all single digit numbers 1, 2, 3, . . . , 9 are Narcissistic. The numbers

153 = 1³ + 5³ + 3³

370 = 3³ + 7³ + 0³

1634 = 1³ + 6³ + 3³ + 4³

are also Narcissistic. Few more examples of Narcissistic numbers are 371, 407, 8208, 9474, 54748. 

Cullen Number is a number that can be expressed in the form 2ⁿ x n + 1.

Few example of Cullen numbers are

• 3 = 2¹ x 1 + 1
• 25 = 2³ x 3 + 1
• 65 = 2⁴ x 4 + 1

Few more examples are 9, 161, 385. 

Munchausen Number is a natural number that is equal to the sum of its digits each raised to the power of itself. Few examples are

1 = 1¹

3435 = 3³ + 4⁴ + 3³ + 5⁵

Factorian is an integer that is equal to the sum of factorials of its digits.

There are exactly four such numbers. They are

1! = 1

2! = 2

1! + 4! + 5! = 145

4! + 0! + 5! + 8! + 5! = 40585.

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