Palindrome Number

Palindromic Number:- Numbers which remains the same when read from the left to right or vice-versa. Indian mathematician Mahavira has mentioned about this in his book.
e.g. 15151, 11000011000011.

If you want to make any number Palindromic here is the simple way :-
Take a number 19
Reverse its digits + 91

Sum it 110
Reverse the digit + 011

Sum it 121

The process will be repeated until you get a palindromic number.

There are several Palindromes number. Mahavira called it पुष्प माल संख्या (garland number).

(Read about Palindrome Primes at -http://www.magic-squares.net/primes.htm)

There are several magic squares formed with the help of Palindromes number.

(Acknowledgement :- I acknowledge the effort by the above mentioned websites for their contribution in the field of Mathematics.I have tried to put their websites for further reference)

Dr Rajesh Kumar Thakru

Friend Or Amicable Numbers

                                         Amicable /Friend Numbers

Two numbers are called Amicable if each of which is equal to the sum of all the exact divisors of the other except the number itself. 220 and 284 are the first pair of amicable numbers.

220 has the exact divisors 1,2,4,5,10,11,20,22,44,55 and 110 whose sum is 284 

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

and 284 has the exact divisors 1, 2, 4, 71, and 142 whose sum is 220. 

1 + 2 + 4 + 71 + 142 = 220

Another pair of amicable number is 1184 and 1210.

The set of 220 and 284 was the first known set of amicable numbers. Pythagoras discovered the relationship and coined the term Amicable because he considered the numbers to be a symbol of friendship.

There is an unauthenticated story related to Amicable numbers that a prince whose name from the stand point of numerology was equivalent to 284 sought a bride whose name would represent 220 believing the fact that such combination would guarantee for a happy marriage throughout the life.

Here is the list of first ten Amicable Pairs:-


1 ---  (220 ,284)
2 --- (1184 , 1210)
3 --- (2620 , 2924)
4 ---  (5020 ,5564 )
5 ---  (6232 ,6368)
6 ---  (10744 ,10856)
7 -- (12285, 14595)
8 -- (17296 , 18416)
9 -- (63020 , 76048)
10 -- (66928 , 66992)

An Arab Mathematician tried to generalize a rule to find the Amicable Number but he could not succed completely. The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the ninth century by the Arab mathematician Thābit ibn Qurra.

It states that if

where n > 1 is an integer and p, q, and r are prime numbers, then 2^n×p×q and 2^n ×r are a pair of amicable numbers.

Dr Rajesh Kumar Thakur

Beast Number

Beast Number:-

666 is called the Beast number. This is termed as unpleasant number. The direct reference of the number 666 can be found in the last book of Bible ‘Revelation’ in chapter 13, verse 18.

It can be written in the following ways:-

666 = 1 + 2 + 3 + 4 + 567 + 89
666 = 123 + 456 + 78 + 9
666 = 9 + 87 + 6 + 543 + 21

In war times this number becomes the handy tools in the hands of the propagandists to ascribe these letters to their opponents by assigning suitable letters pf the alphabet. During Second World War Hitler was ascribed with the title of beast by assigning each letter of English alphabet to a consecutive whole number beginning from 100 for A.

H 107
I 108
T 119
L 111
E 104
R 117

HITLER = 666

The beast number can be written with the help of sixth power of first three natural numbers
666 = 1^6 - 2^6 + 3^6

It can be written as the sum of squares of first seven primes

Dr Rajesh Kumar Thakur

Prime Number

Prime number-
An integer p which is not 0 or ± 1 and is divisible by no integer except ±1 and itself is called Prime number. Donzager stated “Upon looking at prime numbers one has the feeling of being in presence of one the inexplicable phenomena one site of creation.”

2 is the only prime number. There is no perfect technique which can tell us immediately the numbers of prime between two numbers. Though Erastothenes, a great Greek mathematician suggested a method to find the primes between two numbers called Sieve of Erastothenes.

Erastothenes(276-195BC) gave a golden rule though simple it is time consuming which states “First write down the number from 2 to N. Remove all the multiples of 2, 3 and continue the process until all the multiple of primes not greater than √N has been removed.”

Suppose we have to find the primes below 30, first we find the square root of 30 which is 5.477.so we need to remove the entire multiple up to primes 5.


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Hence the primes below 30 are 2,3,5,7,11,13,17,19,23 and,29.


Properties of Primes:-
i) Every natural number greater than 1 has at least one prime.

ii) For every integer a, a^p –a is always divisible by prime p

p 2 2 2 2 3 3 3 3 5 5 5 5
a 2 3 4 5 2 3 4 5 2 3 4 5
a^p-a 2 6 12 20 6 24 60 120 30 240 1020 3120

iii) If N is a prime number then 1+ (N-1)! Is always divisible by N. This is called Wilson theorem.
For N = 2, 1+ (2-1)! = 2 us divisible by 2.
iv) Every odd integers is the sum of a prime and a power of 2, this was claimed in 1848 by De Polignac.
55 = 47+23

v) Every even number except 2 is the sum of two prime numbers.
e.g. 8 = 3 + 5, 16 = 13 + 3 ,60 = 13 + 47 etc. This is called Goldbach conjecture.

vi) Every even integer greater than 4 can be written as the sum of two odd prime numbers. 4 = 2 + 2 = 1 + 3, 6 = 3 + 3 = 1 + 5 …

vii) There is always at least one prime number between n and 2n-2 provided n is greater than 3. If n = 4, 2n-2 = 6 then obviously 5 lies in between 4 and 6. This conjecture was stated by Bertrand (1822-1903).

Interesting Facts

1. A pair of prime numbers is said to be a twin prime pair if the two numbers differ by any 2.
i.e. (3,5)(5,7)(11,13)(17,19)(29,31)(41,43)(59,61)(71,73)etc.
All the twin primes are of the form 6n-1, 6n+1.

2. Between 9,999,900 to 10, 000, 000, there are only 9 prime numbers.
9,999,901; 9,999,907; 9,999,929; 9,999,931 ;9,999,937 ;9,999,943 ;9,999,971 ;9,999,973 ;9,999,991.
But in the next 100 integer from 10,000,000 to 10, 000,100 there are only two primes 10, 000, 019 and 10,000,079.

3. (p,p+2,p+4)is called prime triplet if all numbers are primes

4. The largest known prime number is of 6, 320, 430, digits and was found by Michael Shafer in Dec 2003. It would need 1400 to 1500 pages to write.

5. A gap of 803 composite numbers exits between primes 90874329411493 and 90874329412297 which was found in 1989 by J.Yong and A.Poster



Dr. Rajesh Kumar Thakur

Mathematically 100

One Hundred (100)

1. It is the smallest three digit number.

2. It is the sum of first 10 odd numbers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100

3. Boiling point of water is 100^0C

4. It is the sum of 2 consecutive triangular numbers.
45 + 55 = 100

5. It is the sum of 9 consecutive primes.
2 + 3 + 5 + 7 + 11 +13 +17 + 19 + 23 = 100

6. It is a square number.
10^2 = 100

7. It can be expressed as the sum of cube of first four natural numbers.
1^3 + 2^3 + 3^3 + 4^3 = 100

8. It is an abundant number.

9. It is a centered 33 gonal number.

10. It can be expressed as the sum of four pairs of prime numbers.
47 + 53 = 17 + 83 = 3 + 97 = 41 + 59 = 100

11. It is a Harshad number in base 10.

12. It is a Leyland number which can be expressed in the form of a^b + b^a .
100 = 2^6 + 6^2

13. It is the smallest number whose common logarithm is a prime number.
log 10^100 = 2

14. The atomic number of Fermium is 100.

15. A century has 100 years.

16. There is a very interesting puzzle which says – using digits from 1 to 9, and mathematical operators + and - , make 100. There are several solutions to this puzzle few of them are provided here.
98 – 76 + 54 + 3 + 21 =100

9 – 8 + 76 + 54 – 32 + 1 = 100
98 + 7 + 6 – 5 – 4 – 3 + 2 – 1 = 100
9 – 8 + 76 – 5 + 4 + 3 + 21 = 100
98 – 7 – 6 – 5 – 4 + 3 + 21 = 100


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Dr Rajesh Kumar Thakur

Mathematically 98

Ninety Eight (98)

1. It can be expressed as the fourth power of sum of first three natural numbers.
98 = 1^4 + 2^4 + 3^4

2. The atomic number of Californium is 98.

3. It can be expressed as the difference between two odd consecutive cubes.
98 = 53 – 33

4. The reciprocal of 98 = 1/98 on division shows an interesting pattern of GP, though the patterns discontinues but it starts with the power of 2.
1/98 = 0. 01 02 04 08 16 32 65 30 ….
Here the number shown in bold is in GP.

5. It is the lowest number such that the first 5 multiple of it contains the digit 9.
98 x 1 = 98
98 x 2 = 196
98 x 3 = 294
98 x 4 = 392
98 x 5 = 490

6. It is the smallest number which cannot be expressed as the sum of two primes if the first prime is either of 3, 5 or 7. The next such numbers are 122, 124, 126 and 128.

Dr Rajesh Kumar Thakur

Mathematically 97

Ninety Seven (97)

1. It is a 25th prime number and the largest two digit prime number.

2. It is the highest two digit prime number whose cube has no zero.
97^3 = 912673

3. In Georgian calendar there are 97 leap days in every 400 years.

4. It is a 4 dimensional centered cube number.

5. It is a permutable primes. A prime is called permutable if on a given base, the prime always remain a prime despite its digit’s position changed through permutation. It was first studied by H E Richert. 97 and its reverse 79 are primes .The first 10 permutable primes are- 2, 3, 5, 7, 11, 13, 17, 31, 37 and 71.

6. It can be written in the form of – n^4 + (n +1)^4
97 = 2^4 + 3^4

7. The period of reciprocal of 97 = 1/97 is maximum and its length is 96. More interestingly, the first four pairs of its expansion are in Geometric Progression (GP). Alexender Aitken, a professor at Edinburgh University knew it by heart.
1/97 = 0.01 03 09 27 83 50 51 57 52 57 73 19 58 76 28 86 59 79……

8. It is a Proth prime. A Proth prime is written in the form of – k x 2n + 1, where k is an odd positive integer and 2n > k. It is named after the mathematician Francois Proth. The first 10 Proth primes are- 3, 5, 9, 13, 17, 25, 33, 41, 49 and 57.
97 = 3 x 25 + 1

9. The atomic number of Berkelium is 97

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com