300 Mathematically

1. It is a triangular number.

2. It is the sum of first 24 numbers.
1 + 2 + 3 + ---- + 24 = 300

3. It is also the sum of two consecutive primes, called twin primes
300 = 149 + 151

4. It is the sum of 10 consecutive primes
13+ 17+ 19+ 23 + 29 + 31 + 37 + 41 + 43 + 47

5. It is a palindrome number in 3 consecutive  bases
606₇ = 454₈ = 363₉,

6. In Roman numeral system 
300 = CCC

7. It is the product of three primes 
300 = 2^2 x 3 x 5^2

Dr Rajesh Kumar Thakur

263 Mathematically

                                                           263

1. 263 is a Ramanujan prime.

Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.

( https://en.wikipedia.org/wiki/Ramanujan_prime )

2. 263^2=69169 is a strobogrammatic number that appears the same rotated by 180 degree.

3. 263 = 43 + 47 + 53 + 59 + 61

(Sum of 5 Prime numbers).

4.It is a happy number. Read about Happy number in the following link.

https://www.myqbook.com/MathConcept/667/Happy-numbers

5.It is a balanced prime.

263 = 257+ 269

2

https://en.wikipedia.org/wiki/Balanced_prime

6. It is a Chen Prime.

https://en.wikipedia.org/wiki/Chen_prime

Thanks for reading

Dr Rajesh Kumar Thakur

Mathematically 260

                                                                  260

1. It is a composite number.
2. It is the product of 3 three prime numbers-2, 3 and 13                                                                             260 = 2 x 2 x 5 x 13
3. It is the number of days in Mayan Calender
4. It is the magic sum of a 8 by 8 magic square. 
   
5. It is also the sum of Franklin magic square 
   

Dr. Rajesh Kumar Thakur

Source :- 

1. Wikipedia

2. Pats' Blog

3. The Power of Mathematical number

Mathematically 244

1. 244 is an anti-perfect. 

  Its proper divisors are 1, 2, 4, 61, and 122, & adding their reversal is 244.

                           1 + 2 + 4 + 16 + 221 = 244.

2. It is a palindromic in bases 3 (100001₃), 11 (202₁₁), 60 (44₆₀), 121 (22₁₂₁), 243 (11₂₄₃).

3. 244 is also the sum of three cubes,
                                   244=1^3+3^3+6^3 

4. It is the sum of two nonzero fifth powers
                                  244 = 1^5 + 3^5

5. It is a Harshad number on the base 3, 9, 11 ... . Harshad number was discovered by Indian mathematician Kaprekar.

Dr Rajesh Kumar Thakur

Mathematically 240

240

1. It is a semiperfect number.

2. It is a highly composite number since it has 20 divisors total (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, and 240), more than any previous number.

3. It is a refactorable number or tau number, since it has 20 divisors and 20 divides 240.

4. It is a pronic number since it is the product of 2 consecutive number
240 = 15 × 16

5. 240 can be expressed as a sum of consecutive primes in two different ways: 240 = 53 + 59 + 61 + 67
 = 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43.

6. It is the product of 6 Fibonacci number.
240= 1 × 1 × 2 × 3× 5× 8

Number 211

Number 211

30th July in a normal year and 31st July in a leap year is the 211th day of the year
1. It is an odd number

2. It is the sum of three consecutive primes.
211 = 67 + 71 + 73

3. It is a Chen Prime. A prime number p is called a Chen prime if p + 2 is either a prime or product of two primes.

Rajesh Kumar Thakur

Number 210

Number 210

1.  (21, 20, 29) and (35, 12, 37) are the two least primitive Pythagorean triangles with different hypotenuses and the same area (=210).

2. It is the product of first four prime number
                         210 = 2 x 3 x 5 x 7

3. It is the sum of 8 consecutive prime numbers.
                   13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210

4. It is a triangular Number. 
                   190              210                    231

5. It is a Harshad Number. To know more about Harshad Number. Harshad number was discovered by Indian Mathematician Kapekar.

6. It is a Pronic Number. Pronic number is a number which is the product of n and n+ 1
                210 = 14 x 15


7. It is an abundant Number. Abundant number is a number whose sum of proper divisor is more than the number. 
12 is the first abundant number.
Factor of 12 are 1, 2, 3, 4 and 6
1 + 2 + 3 + 4 + 6 > 12

8. It is a composite number, even number and pentagonal number.

Dr Rajesh Kumar Thakur

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