Brahmagupta - Fibonacci Identity

                                                      Brahmagupta - Fibonacci Identity 

There is a very famous identity used in algebra known as – Brahmagupta – Fibonacci identity that expresses the product of two sums of two squares as a sum of two squares in two different ways.

                        (a² + b²) (c² + d²) = (ac – bd)² + (ad + bc)²      

                                                  = ( ac + bd)² + (ad – bc)²

For example:-

(1² + 4²) (2² + 7²) = (2 – 28)² + (7 + 8)²

                            = 26² + 15²

And

(1² + 4²) (2² + 7²) = (2 + 28)² + (7 – 8)²

                            = 30² + 1²

^{Dr Rajesh Kumar Thakur}

Mathematically 101

                                                               101

1. One hundred one is a sexy prime as well as a sexy prime triplet. Sexy primes differ from each other by 6. If p is a prime and p + 6 is also a prime then it is called a sexy prime.
Example:- (5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107)

2. 101 is also a sexy prime triplet. If p, p+6 and p + 12 are all prime then it is called sexy prime triplet.
Example:- (5,11,17), (7,13,19), (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113),

3. It is the sum of 5 consecutive primes
                                       13 + 17 + 19 + 23 + 29 

4. It can be written as the sum of first 5 factorial with alternate sign.
                                  101 = 5! - 4! + 3! - 2! + 1!

5. It is a palindrome number and a palindrome prime.

6. It is the largest prime in form of 10^n + 1
                                   101 = 10^2 + 1

7. There are 101 digits in the product of the 39 successive primes produced by the formula n² + n + 41, where n = 1 to 39. This formula was used by Charles Babbage to demonstrate the capabilities of his Difference Engine (1819-1822))

Happy reading 
Dr Rajesh Kumar Thakur

Self Number

                                                          D R Kaprekar (1905 -1986)
Self number / Devlali Number / Swayambhu' Number was discovered by Indian mathematician Dattathreya Ramchandra Kaprekar who was born in Dhanu (Maharastra). 

A self number is a number that cannot be written as the sum of any other integer n and the individual digits of n. It was discovered in 1949 by Kaprekar. 

These are the self number 
1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244 - - -

Start with a number, say 23. The sum of its digits (2 + 3) are 5 which we add to 23 to obtain 28. Again add 2 and 8 to get 10 which we add to 28 to get 38. Continuing gives the sequence

23, 28, 38, 49, 62, 70, ...

These are all generated by 23. But is 23 generated by a smaller number? Yes, 16 generates 23. In fact the sequence we looked at really starts at 1

1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, ... 

Here is the recurrence formula to find the Self Number

where 

(Sources :- Wikipedia / Mac Tutor Archive )

Send your comments to  -

rkthakur1974@gmail.com

Dr Rajesh Kumar Thakur

Krishnamurthy Number

A number is said to be Krishanmurthy number if the sum of factorial of all digits of a number is equal to the number.

1! = 1.
2! = 2.
1! + 4! + 5! = 1 + 24 + 120 = 145.
4! + 0! + 5! + 8! + 5! = 24 + 1 + 120 + 40320 + 120 = 40585.

There are only 4 such number found so far.

Dr. Rajesh Kumar Thakur

Perfect Number

A number is said to be Perfect if the sum of factors of its proper divisor is equal to the number itself.
Example:- 6, 28, 496, 8128 ...

6 is the smallest Perfect number.
St. Augustine wrote in The City of God (413–426):

Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect.

The first reference of Perfect number is found in Euclid's Element (Volume 9) where Euclid proved that 2^p−1(2^p − 1) is an even perfect number whenever 2^p − 1 is prime

The first four numbers according to this definition are

for p = 2: 2^1(2^2 − 1) = 6
for p = 3: 2^2(2^3 − 1) = 28
for p = 5: 2^4(2^5 − 1) = 496
for p = 7: 2^6(2^7 − 1) = 8128.
Perfect Number can also be expressed in form of Triangular Number

(Source of Image :- Wikipedia)

Dr Rajesh Kumar Thakur

Largest Prime Number M77232917 discovered on January 4 , 2018

A FedEx employee Jonathan Pace ,an engineer by profession has discovered the largest prime Number. According to GIMPS’s (Great Internet Mersenne Prime Search) website, the newly discovered prime number is calculated by raising 2 to the 77,232,917th power and subtracting 1.

M77232917 itself is reportedly 23 million digits long. According to New Scientist, it is one million digits longer than its predecessor, which clocked in at 22 million digits.

The greatest prime number discovered before M77232917 was found in 2015, and was 5 million digits longer than the one that came before it in 2013.

Although Euclid proved that if 2^P-1 is prime, then 2^P-1*(2^P-1) is a perfect number in 350 BC, the French monk Marin Mersenne was honored with the name for his conjecture of which prime numbers could be used for P to produce larger primes. Although written in the early 17th Century, the conjecture took 300 years to prove. Meanwhile, Euler also got in on the act, proving that all even perfect numbers are formed this way.

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form of 2^p − 1 for some integer p.

Dr Rajesh Kumar Thakur

Friedman Number

A Friedman number is a positive integer which can be written with the help of symbols like - + , - , x, / , (), ^ etc. and using the digits of the numbers.

Example:-
25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024,
1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349,
2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592 ,2737, 2916, 3125,
3159, 3281, 3375, 3378, 3685, 3784, 3864, 3972, 4088, 4096, 4106, 4167, 4536, 4624,
4628, 5120, 5776, 5832, 6144, 6145, 6455, 6880, 7928, 8092, 8192, 9025, 9216, 9261.

Explanation:-
25 = 5^2
121 = 11^2
125 = (5)^1 + 2 = 5 ^3
126 = 21 x 6
127 = -1 + 2^7
289 = (8 + 9)^2
343 = (3 + 4 ) ^3

Friedman numbers are named after Erich Friedman, an Associate Professor of Mathematics in Florida in US.


Enjoy
Rajesh Thakur

References:-
1. https://en.wikipedia.org/wiki/Friedman_number
2. http://www2.stetson.edu/~efriedma/mathmagic/0800.html