devotion & mathematics

The Euler Totient Function Ჶ(n) :                                                      If n ≥ 1 , the Euler Totient Function Ჶ(n) is defined to              be the number of positive integers not exceeding n which are relatively prime to             n ;  thus ,                         Ჶ ( n ) = 𝚺 1 for 1 ≤ k ≤ n, here 𝚺 indicates that the sum is extended over                                                                       those k relatively prime to n.  For Example :      Suppose n = 1.       ...

Number theory , like many other branches of mathematics , is often concerned  with sequences of real or complex numbers. In number theory such sequences are called arithmetic functions. Definition :                    A real or complex - valued functions defined on  the positive integers is called an arithmetical function or a number- theoretic function. The Mobius Function μ (n) :                    The Mobius function μ is defined as follows :                         μ(1) = 1 ;                     If n > 1 , write n = p 1 a1 + p 2 a2 + … + p k ak . Then                                           μ( n) = ( -1 ) k if a 1 = a 2 = … = a k =1     ...

                                  Greatest Common Divisor for more than two numbers :                                  The greatest common divisor of three numbers ab,c is denoted by    ( a, b, c ) and is defined by gcd { a,b,c } = (a,b,c)                                                                     = ( a, ( b,c) )                                                                     = ( (a,b) , c).                i.e g.c.d depe...

  Euclid Algorithm or Division Algorithm :  Statement :               Given integers a and b with b > 0 , there exists a unique pair of integers q and r such that               a = bq + r , with 0 ≤ r < b. Moreover , r = 0 if, and  only, if b/a. Proof :              Suppose a and b are two integers with b > 0.               Let S = { y / y= a - bx , x is an integer, y ≥ 0 } .              Clearly S is a non - empty set of non - negative integers.              ∴ By well- ordering Principle , S must have a smallest member , say a - bq for                   some integer q .              Let r = a - bq.              Hence r is the sm...

  The Fundamental Theorem of Arithmetic :    Statement : Every integer n > 1 can be represented as a product of prime factors in                            only one way, apart  from the order of the factors.  Proof :      Consider the statement       P(n) : Every integer n >   1 can be represented as a product of prime factors in only                  one way.       Now we prove this statement by using mathematical induction on n.        We have 2 = 2 1                            = A product of prime factor.       The statement is true for n=2 i.e. p(2) is true.       Ass...