*The theory of numbers, more than any other branch of pure mathematics, has begun by being an empirical science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they have been proved; and they have been suggested by the evidence of a mass of computation.*

**G. H. Hardy**

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Welcome to the world of numbers, where the secrets of mathematics come alive in fascinating and unexpected ways. Numbers, the building blocks of our numerical system, hold a universe of intriguing facts and captivating stories. In this blog post, we invite you on a journey through **40 Fascinating Facts about Numbers**, where we will uncover the hidden wonders and peculiarities that numbers conceal.

This exploration will ignite your imagination, challenge misconceptions, and reveal the remarkable intricacies of the numerical realm. Whether a math enthusiast or simply curious, there’s something for everyone in this captivating journey. Let’s marvel at the beauty embedded within numbers.

So, get ready to embark on an adventure that will transform the way you perceive numbers. Whether you’re a mathematician or a curious explorer, **40 Fascinating Facts about Numbers** promises to be an enlightening and enchanting experience.

To begin with, let’s explore the intriguing properties of the perfect square number **1089**. This number can be obtained by taking the difference between the squares of two numbers that are formed by reversing the digits of each other:

1089 = 65

^{2}– 56^{2}.

Fascinating, isn’t it? Could you provide more instances?

2! (= 2) is a factorial that is prime. Do you have any other illustrations in mind?

The fear of the number 13 is called *triskaidekaphobia*.

The numbers 139 and 149 are the smallest consecutive prime numbers whose difference is 10. Can you provide more examples?

If we take the product of the digits of the number 19 and add it to the sum of these digits, we will end up with the number 19. Thus,

19 = (1 × 9) + (1 + 9).

Can you find other such numbers where this is true?

There are only 10 types of people in the world . . .

Those who understand binary and those who don’t.

23 is a prime number with consecutive digits. Can you provide more examples?

The natural number 2 satisfy the relation

2 + 2 = 2 × 2.

Do you know a different example?

Given natural numbers 3 and 4. Which (mathematical) symbol can be placed between 3 and 4 so that the result will be a number greater than 3 and less than 4?

Surprisingly, a decimal point symbol can solve the problem (and the number in this case is 3.4).

The number 85 can be expressed as the sum of two squares in two different ways. In fact,

85 = 9

^{2}+ 2^{2}85 = 6

^{2}+ 7^{2}.

Are you aware of any other examples?

The sum of all the factors of the square number 81 is equal to another square number 121. In fact, we have

1 + 3 + 9 + 27 + 81

= 121

= 11

^{2}.

Do you have any other illustrations in mind?

The number 999999937 is a prime and it is the largest prime number less than 1 billion.

When the number 497 is doubled, one gets the reversal of the number 497 increased by 2. Symbolically,

497 × 2 = 994, and

497 + 2 = 499,

the reversal. Do you know any other examples?

The product of all the factors 48 is surprisingly 48^{5}. In fact, all the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 and the required product is

1 × 2 × 3 × 4 × 6 × 8 × 12 × 16 × 24 × 48

= 254803968

= 48

^{5}.

There are 7 letters in TUESDAY, which is the only day of the week whose name contains a prime number of letters.

2592 = 2

^{5}× 9^{2}

is the only number of this kind. This is known as the* printer’s error*. Do you know any other number satisfying the relation

x×^{y}u=^{x}xyux?

121 is not a prime number, regardless of the base we choose. In fact, in base *a* we have

121

= 1 ×

a^{2}+ 2 ×a+ 1= (

a+ 1)^{2}= (

a+ 1)(a+ 1),

a composite number.

Are you aware of any other examples?

Choose a prime number greater than 3. Multiply it by itself and add 14. If the result is divided by 12, then the remainder will always be 3.

The number 6534 is indeed peculiar. It is equal to 3/2 times of its reversal. Symbolically,

6534 = (3/2) × 4356.

Can you provide more examples?

4567 is a four-digit prime with consecutive increasing digits.

Are you aware of any other examples?

We know that

30 ‒ 33 = ‒3.

How can we move one digit so that the answer becomes 3? The answer is rather simple:

30 ‒ 3

^{3}= 30 ‒ 27

= 3.

The number 3435 is an unusual number, since it can be expressed as the sum of its digits each taken to the power equal to the digit. Symbolically, that is

3435 = 3

^{3}+ 4^{4}+ 3^{3}+ 5^{5}.

Can you find any other number satisfying this property?

1, 2 and 3 are three consecutive natural numbers such that their sum and product are equal. In fact

1 + 2 + 3 = 1 × 2 × 3.

Can you find a different set of such numbers?

The number 31 has a curious property that can be best seen symbolically:

31 = 1 + 5 + 5

^{2}31 = 1 + 2 + 2

^{2}+ 2^{3}+ 2^{4}.

The number 135 can be expressed as the sum of the increasing powers of its digits:

135 = 1

^{1}+ 3^{2}+ 5^{3}.

Do you know any other number with this property?

The number 264 shares the amusement, as it is equal to the sum of different two-digit numbers that can be formed from it. Symbolically,

264 = 26 + 62 + 24 + 42 + 64 + 46.

Are you aware of any other examples?

8 is the largest cube in the Fibonacci sequence (1, 1, 2, 3, 5, **8**, 11, 19, . . .).

The number 1634 can be represented as the sum of each of its digits raised to the (fixed) fourth power. Symbolically, it is

1634 = 1

^{4}+ 6^{4}+ 3^{4}+ 4^{4}.

Can you find any other examples?

Another beautiful number is 113. This is a prime. Even more surprising is that, all possible arrangements of this number are also primes. In fact, rearrangement of the digits leads in the primes 131 and 311.

1026753849= 32043

^{2}

is the smallest perfect square number with all the digits (used exactly once) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Like **12** and **21** are reverse of each other, so is their squares **144 (= 12 ^{2})** and

**441 (= 21**.

^{2})Can you find more examples of this?

220 and 284 are the numbers made for each other in the way that the sum of the factors of one number is equal to the other number.

Any other examples?

When we write 666 using Roman symbols, it uses all the numerals in descending order:

666=DCLXVI.

Consider the numbers 6, 480 and 495. **Sum** of these numbers is **981** and their **product** is **1425600**. Consider one more set of numbers 11, 160 and 810. Their sum and product will remain invariant as above. Can you find one more example?

**7** is a natural number followed by a perfect cube (= **8**) and a perfect square (= **9**). Can you find a different one?

The square

69

^{2}= 4761

and the cube

69

^{3}= 328509

of the number **69** uses all the decimal digits exactly once.

The number 48 have a very peculiar property. If we add 1 to 48, we get the square number 49. If 1 is added to the half of 48 (i.e., 24), we also get square number 25. We can write

48 + 1

= 49

= 7

^{2}

48 ÷ 2

= 24; and

24 + 1

= 25

= 5

^{2}.

Can you find other number(s) that share this peculiar property?

The Hardy-Ramanujan number have so many beautiful properties. One of them is it can be written as the product of two reversed numbers:

1729 = 19 × 91.

Can you find any other number having this property?

There are pairs of triplet-numbers whose products all use the same digits in different orders. For example,

333 × 777 = 258741

333 × 444 = 147852.

Can you find one more examples?

**26** is the natural number sandwiched between a *perfect square* (= **25**) and a *perfect cube* (= **27**).

Can you find a different one?

This blog is as much yours as it is mine. So, if you have got some ideas to share what you want to see in the next post, feel free to drop a line. We welcome your ideas with open arms and reverence! Looking forward to seeing you soon on “**Math1089 **–** Mathematics for All**” for another fascinating mathematics blog.