Amicable Number

The two digits of the number of A and B will be friendly to each other if

sum of all proper manufacturers of  A  and sum of all proper 

manufacturers  of B  are  equal. So thus two digits are called 

Amicable Number.

Such as 220 and 282 are Amicable Number . Because

The manufacturers of 220 are 1,2,4,5,10,11,21,44,53 and 110.

The sum of manufacture is (1+2+4+5+10+11+21+44+53=282).

Besides the  manufacturers of 282 are 1,2,4,71 and 142.

 The sum of manufactures of (1+2+4+71+142=240).

so 220 and 282 ar friendly to each other .

Ten amicable number are (1184,1210),(2620,2924),(5020,5564),

(6232,6368),(10744,10856),(12285,14595),(17296,18416),

(63020,76084).

Amicable Number pair was discovered by the Pythagoreans .

A few interesting characters :

1.  No amicable number thats one of two numbers is a square.
   
2. Some amicable number (x,y) in which the sum of digits of x and y is equal . Such as (100485,124155)
   100485=1+0+0+4+8+5=18124155=1+2+4+1+5+5=18  

Repdigits and repunits

A repdigit is a natural number with one repeating digit; the name, in fact, comes from the term “repeated digit.” The most famous redigit is the so-called “Beast Number” 666, a common symbol of the antichrist or of Satan. A repunit, then, is a repdigit that only uses the number 1; repunits pop up frequently in binary code and are related to that most famous of primes, Mersenne Primes. It has been conjectured that there are an infinite number of repunit primes, so if you’d like to try to prove it, please do so at your leisure.

Narcissistic numbers

Narcissistic numbers, also known as Armstrong numbers or “pluperfect digital invariants,” are numbers that—listen closely—are equal to the sum of each of its digits when those digits are raised to the power of the AMOUNT of digits in the number.

Ok. What? Let’s take an example of the four existing narcissistic cubes:

153 = 1^3 + 5^3 + 3^3
370 = 3^3 + 7^3 + 0^3
371 = 3^3 + 7^3 + 1^3
407 = 4^3 + 0^3 + 7^3

In these cases, each digit is cubed because there are three digits in the number. Then, those cubed numbers are added together to produce a sum equal to the original number. There are no 1-digit narcissistic numbers, nor 12 or 13-digit ones; the two 39-digit ones are:

115132219018763992565095597973971522400 and 115132219018763992565095597973971522401.

English mathematician G. H. Hardy recognized the frivolity of such numbers by proclaiming in his book “The Mathematician’s Apology” that “These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.”

Happy Numbers

Some numbers are weird; others are happy. If you’d like to find out if a given number is happy, you’ll need to perform the following set of operations. Let’s take the number 44:

First, square each digit, then add them together:

4^2 + 4^2 = 16 + 16 = 32

Then, we’ll do it again with our new number:

3^2 + 2^2 = 9 + 4 = 13

And again:

1^2 + 3^2 = 1 + 9 = 10

And finally:

1^2 + 0^2 = 1 + 0 = 1

Voila! It’s a happy number. Anytime you take a number, perform this “procedure,” and eventually arrive at the number 1, you have yourself a happy number. If your number never reaches 1, then sadly, it’s unhappy. Interestingly, happy number are extremely common; there are 11 of them between 1 and 50, for example.

As a final note, the greatest happy number with no recurring digits is 986,543,210. That is a happy number indeed.

Untouchable numbers

While weird numbers are not equal to the sum of any of their divisors,untouchable numbers take it a step further. For a number to be untouchable, it must not be equal to the sum of the proper divisors of ANY number. A few untouchables are 2, 5, 52, and 88; in fact, 5 is thought to be the only odd untouchable number in existence (though it hasn’t been formally proven). There are an infinite number of untouchable numbers, meaning there is no such thing as the largest one.

Weird numbers

What are weird numbers? To understand them, we must first begin with “abundant” numbers. Abundant numbers, also known as “excessive,” are bigger than the sum of their proper divisors. 12, for instance, is the first (smallest) abundant number—the sum of its proper divisors, 1+2+3+4+6, is 16. 12, therefore, has an “abundance” of 4, the amount by which the sum of its divisors exceeds the number. There are many even abundant numbers, but we don’t get to an odd one until the number 945.

Some abundant numbers are “semiperfect” or “pseudoperfect,” meaning that they are equal to all or just some of their proper divisors. 12 is an imperfect abundant number because some of its divisors can be added together to form 12.

At last, we arrive at weird numbers. A number is weird if it is abundant but NOT semiperfect; in other words, the sum of its divisors is larger than the number itself, but no subset of divisor sums equal the number. Weird numbers are uncommon – the first few are 70, 836, 4,030, and 5,830.

Powerful numbers

Achilles was a powerful Trojan War hero who was extremely powerful but had one flaw—his achilles heel. Like him, Achilles numbers are powerful but not perfect.

So, let’s begin with a powerful number. A number is considered powerful if all of its prime factors remain factors once they are squared. 25 is a powerful number because its one prime factor, 5, remains a factor once its been squared (25, which goes into 25 once). Now let’s move onto perfect powers, number that can be expressed as an integer power of another integer; 8 is a perfect power, as it’s 2 cubed.

So now, back to the original premise – Achilles numbers are powerful, but they are not perfect powers. 72 is the first Achilles number, as it is powerful, but it is not a perfect prime. Others include 108, 200, 288, 392, 432, 500, and 648.

Interesting numbers

There is an old paradox in the world of mathematics that is known as the “interesting number paradox.” Simply put, if you keep counting natural numbers, eventually you’ll encounter one that isn’t interesting; where it gets paradoxical is that by virtue of being the smallest uninteresting number, that number has now become interesting.

Of course, this is all subjective, as it relies on a vague definition of the word “interesting.” Very generally speaking, a number is considered interesting if it has some type of mathematical quality that sets it apart; 19 is interesting because it’s prime, 999 is interesting because it’s a palindrome (and the UK version of 911); 24 is interesting because (among other reasons) it’s the largest number divisible by all numbers less than its square root. Mathematicians

Emirp

“Emirp” is the word “prime” spelled backwards, and it refers to a prime number that becomes a new prime number when you reverse its digits. Emirps do not include palindromic primes (like 151 or 787), nor 1-digit primes like 7. The first few emirps are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, and 157 – reverse them and you’ve got a new prime number on your hands.

Understanding the most beautiful equation in Mathematics

 Euler was one of the most influential and prolific mathematicians in history. He had published over 800 papers and 20 books, making him the greatest contributor in mathematics. Referred as the Mozart of Mathematics, Euler left hardly any area of Mathematics untouched, contributing to various field like mathematical analysis, number theory, mechanics and hydrodynamics, cartography, fluid dynamics and topology. In this article, we'll try to understand the most beautiful equation in all of mathematics:

eiπ+1=0

It connects the five most important constants of mathematics and three most important mathematical operations - addition, multiplication and exponentiation. So, how did Euler arrived at this result?

The Euler's constant e is defined as lim(1+1n)nas n approaches infinity.It's approximate value is equal to 2.71828. In his most influential work, Introductio in analysin infinitorum, Euler defined the function e^x in analysis as: ex=lim(1+xn)n as n tends to infinity. So, we get: ex=1+x1!+x22!+x33!+...

This is the known series for ex$e^x$ Euler's brilliant mathematical mind replaced the real variable x with ix were i = −1−−−√$\sqrt{-1}$ .

So, we get: eix=1+ix1!+(ix)22!+(ix)33!+..

We know that square of i is equal to -1. So, replacing subsequent values for i3,i4,i5...$i^3,i^4,i^5...$ , we get:

eix=1+ix1!−x22!−ix33!+.

On separating real and imaginary parts, we have: eix=(1−x22!+x44!+...)+i(x−x33!+x55!+...)${\mathrm{<br>}}^{}$

So, we got two trigonometric series of cosx and sinx respectively. Hence, eix=cosx+isinx

If we put, x=π$x=\pi$ , we get:

eiπ=−1ascosπ=−1

and sinπ=0

or

eiπ+1=0

So, this was the story of the creation of most beautiful equation in mathematics. I would like to end this article by a quote from professor Keith Devlin - "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."