Number in the Name of Mathematicians

Numbers Bearing the Names of Mathematicians

Were it not for number and its nature, nothing that exists would be clear to anybody either in itself or in relation to other things. You can observe the Power of numbers exercising itself not only in the affairs of the demons and gods but in all the acts and thought of men, in all handicraft and music.

Everybody loves to play with numbers. Have you played with numbers? While playing snakes and ladders you certainly play with numbers. There are several mathematicians who got fascinated with the charms of numbers and this section of chapter deals only with those numbers which has been named after its discoverer. These numbers have special features and they are special because they bear the name of mathematicians. For the convenient of the readers a brief Bio-Data has been provided here so that they can get the right information without much effort.

1. Ramanujan Number----   1729 is called the Ramanujan Number. This is the least number which can be easily expressed as the sum of the cubes of two numbers in different ways. It can be written as 10³+9³ and 12³+1³. This was in fact the number of a taxi Prof Hardy had hired when he came to see ailing Ramanujan in the hospital. When Prof Hardy asked Ramanujan that the taxi he had hired had a boring number; Ramanujan instantly said that this is the smallest number which can be expressed as the sum of cubes of two numbers in two different ways.

Ramanujan was born on December 22, 1887 at Erode in Madras Presidency in a very poor family. He passed his primary education in 1897 scoring first in district but could not pass his F.A. Prof Hardy was very much impressed when he got a letter from Ramanujan and invited him England. He was awarded the BA degree and he became the Fellow of Royal Society. Prof Hardy wrote “Ramanujan was my discovery. I did not invent him- like other great men, he invented himself- but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognize at once what treasure I had found.” He worked on partition numbers, hyper geometric series, and divergent series.

 

                                    

  

2. Euler Number:-This is the smallest number which can be expressed as the sum of two fourth power in two different ways. This special number was presented by great mathematical genius Leonhard Euler (1707-1783).

                                    635318657= 133⁴ + 134⁴ = 59⁴ + 158⁴

Leonahrd Euler was born on 15 April 1707  in Basel in Switzerland. He was an exceptionally brilliant mathematician. He completed his Master’s degree in Philosophy in 1723.  He served as a medical Lieutenant in the Russian Navy from 1727 to 1730. In 1730 he became a Professor of Physics at the Academy and left the Russian Navy Port job. Tragically when he was at the age of 28 he became completely blind but this true mathematics lover never took a rest. While blind he published more than 400 mathematical papers, most of which he dedicated to his servant untrained in mathematics. He is credited for the development for a new branch of mathematics called Topology. On 18 September 1783 he died in St Petersburg in Russia.

3. Harshad Number:-  A number is called Harshad Number if it is divisible by the sum of its digits. 1729 is a Harshad number as it is divisible by the sum of its digits 1 +7 + 2 + 9 = 19. The word Harshad has its origin from the Sanskrit language and it means great joy. Harshad Numbers were defined by the Indian mathematician D.R.Kaprekar. It is also known as Niven numbers in the name of mathematician Ivan M. Niven.

4.. Kaprekar Number:-   6174 is called Kaprekar number.

 Here is the method to find a Kaprekar Number:-

Take any four digit number whose all digits are not equal. Arrange this number in descending order and then reverse it. Subtract these two. Repeat the process after few steps once or a while you shall find 6174.

D R Kaprekar was a famous mathematician of India. He was born on Jan 17, 1905 at Dhanu near Bombay. In 1946 he had discovered a constant known as Kaprekar constant or Kaprekar Number. He is also credited to have discovered Harshad Numbers.

 

Take a number                                                             1234

Arrange in descending order                                      4321

Reverse it                                                                    1234

----------------------------------------                                              ----------

Subtract it                                                                   3087

Arrange in descending order                                      8730

Reverse it                                                                    0378

----------------------------------------                                   ---------

Subtract it                                                                   8352

Arrange in descending order                                      8532

Reverse it                                                                    2358

----------------------------------------                                               ----------

Subtract it                                                                   6174(Kaprekar constant)

Moreover, there is another form of Kaprekar number. A number is said to be Kaprekar number if for a given base the sum of square of the two split part of the number gives back the original number. Example :- 99 is a Kaprekar number                                                                                                        99² = 9801 and 98 + 01 = 99

The first ten Kaprekar numbers are -1, 9, 45, 55, 99, 297, 703, 999,2223 and 2728

5. Fermat Number: - Numbers of the type   Fn =    

for n =1, 2, 3… are called Fermat Numbers. Piere De Fermat wrote to Mersenne on December 25, 1640 that:

If I can determine the basic reason why 3, 5, 17, 257, 65537... are prime numbers, I feel that I would find very interesting results, for I have already found marvelous things(along these lines) which I will tell you about later.    

Fermat thought that these numbers might be primes but it is not true for F5. In 1732 Euler discovered that F5 is divisible by 641.

F5 = 641 x 6700417 =4294967297

Pierre de Fermat (1601-1665) was a brilliant and versatile French amateur mathematician. He is considered as the founder of probability theory along with Pascal. He was also considered the co-inventor of Analytic Geometry along with Rene Descartes

                                                                    

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6. Fibonacci Number: - 1, 1, 2, 3, 5, 8, 13… are the Fibonacci numbers. It is the set of numbers which after the second is the sum of the two previous ones.1+1=2, 1+2=3, 2+3 =5 and so on. The definition of the Fibonacci series is:-

F_n+1 = F_n-1 + F_n,  if n> 1

                                                F₀ = 0 and F₁ = 1

Fibonacci or Leonardo Pisano was born in Pisa in 1170 but was educated in North Africa where his father was working. He had written many books but Liber Abaci, Liber quadratorum,  Practica Geometriae are still available. Liber Abaci a book published in 1202 was based on Arithmetic and Algebra. This book introduced the European with the Hindu- Arabic Numerals and place value system. Fibonacci number is also discussed in Liber Abaci. He died in 1250 probably in Pisa.

                                                                       

Fibonacci numbers can be generated with the help of the Pascal’s triangle.

In general the Pascal’s triangle is written as

	1                      1     1                              1    2    1                      1    3    3    1                           1    4   6   4   1                      1   5   10    10   5    1                      1  6    15     20   15  6   1

 

Now in order to find the Fibonacci numbers from it arrange its digits so that 1’s in the left hand are below each other as shown in the figure below. Fibonacci numbers can be obtained by summing up the arrowed numbers as shown here.

7. Liouville Number  Any number of the form

     

where A is a constant, is called a Liouville Number

Liouville Joseph(1809-1882) was a French analyst and geometer. He was the first to prove the existence of transcendental numbers. In 1844 he proved that transcendental numbers existed by actually constructing examples. He also proved that such numbers cannot be the solution of polynomial equation with integer coefficients and hence are not algebraic.

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8. Pythagorean Numbers:- Any set of positive integers satisfying the equation

                                                x² + y² = z²

    is called Pythagorean numbers. Such numbers are given by

       m²- n² , 2mn,  m² + n² where m and n are arbitrary positive integers with the condition that m ≠n . More over   n, ^½(n²-1) and ^½(n²-1) +1 are also the set of Pythagorean triplets.

( 3,4,5), (5,12,13), (6,8,10), (7,24,25),(8,15,17),(9,40,41),(11,60,61)… are the example of Pythagorean numbers.

Pythagoras of Samos (569BC-475BC), a great Greek mathematician is famous for his theorem better known as Pythagoras Theorem which states” the sum of the square of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.” He founded a school in Samos, his birthplace, which was called Semicircle. He made remarkable contribution   in music, mathematics and astronomy. Numbers were his personal friend. He used to play with pebbles forming a group of different numbers. The pebbles made it possible for Pythagoreans to identify different shapes, the simplest being the two dimensional figure the triangle and simplest three dimensional figures was the tetrahedron. Aristotle in his Metaphysics writes “They (the Pythagoreans) supposed the elements of numbers to be the elements of all things and the whole heaven to be a musical scale and a number ...Evidently then these thinkers also consider that number as the principal both as matter for things and as forming both their modification end their permanent states.”

  

9. Markov Number: - It is a positive integer x, y, z that is the part of the solution of the equation           

                       

The first ten Markov numbers are – 1, 2, 5, 13, 29, 34, 89, 169, 233 and 433. It was discovered by Andrey Markov.

10. Smith Number: It is a composite number. For a given base, the sum of its digits is equal to the sum of the digits in its prime factorization.                                                                                                         Example: - 22 is a smith number because it has two prime factors 2 and 11 and the sum of 22 is equal to the sum of digits of its prime factors 2 and 11--                                                                   2 + 2 = 2 + 1 + 1

This number is named after Harold Smith and was studied by Albert Wilansky of Lehigh University. Albert notices this property in the phone number of his brother in law Harold and dedicated such number by giving his name. The phone number of Harold smith was 4937775. The prime factorization of 493775 is 3, 5, 5, and 65837.                                                                                   4937775 = 3 × 5 × 5 × 65837 and                                                      4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42.

The first ten Smith numbers are -- 4, 22, 27, 58, 85, 94, 121, 166, 202 and 265.

11. Sohie Germain Prime numbers: - It is a prime number. For a given prime p, if 2p + 1 is also a prime it is called Sophie Germain Prime.  It is named after French mathematician Marie Sophie Germain. The first few such numbers are –2 , 3, 5, 11, 23 ,29, 41..

12. Lucas Numbers:- A series of numbers

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521…

is called Lucas numbers

L_n = L_n-1 + L_n-2

 for n>1 is called Lucas series where L0 = 2 and L1 = 1.

Edouard Lucas(1842-1891 )a French mathematician is best known for his result in number theory. He studied the Fibonacci sequence and found that if the Fibonacci rule of adding the last two to get the next is kept and the series is started from 2 and 1 instead of 0 and 1 a new series 2, 1 , 3, 4, 7, 11, 18,… is formed. This series is called Lucas series in his honour. Lucas is also known for the invention of the Tower of Hanoi puzzle and other mathematical recreation.

 

13. Mersenne Numbers:-Numbers in the form of  M_n = 2^p – 1 are called Mersenne Numbers. In 1644 Mersenne claimed that if p  = 2, 3 , 5, 7, 13, 17, 19, 31, 67, 127, and 257 ; Mn  is prime and for all the other n<257  Mn is composite. This conjecture was proved wrong over the years.

Marin Mersenne (1588-1648) was a Monk in a Church in France. He investigated the prime numbers and worked on finding formulae that would represent all the primes. Though he failed to do so but in doing so he conjectured that in  Mn = 2n – 1 , if n = 2, 3, 5, 7, 13, 17, 19, 31, 67, and 257 the Mersenne Numbers would be Prime.

                                   

     

           

14. Wison Prime:- If p is a prime such that p² divides (p – 1 )! + 1 it is called Wilson number, where ! denotes the factorial of a number. It is named after English mathematician John Wilson (1741 – 1793). The only known Wilson primes are 5, 13 and 563

There are many such number which is not addressed here due to some limitation and I do hope that the readers after enjoying such numbers will take initiative to find out such numbers themselves.     

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Rajesh kumar Thakur

rkthakur1974@gmail.com