devotion & mathematics

Fermat's Theorem : Statement :                       If  p is a prime and (a,p) = 1 then a p-1 ≡ 1 (mod p). Proof :                Suppose p is a prime number and (a,p) = 1.         Since (a,p) = 1 then the numbers a,2a,3a,…,(p-1) are divided by p and the remainders are 1,2,3 …,                 (p-1) ; not necessarily in this order.          Let a ≡ r 1 ( mod p), 2a ≡ r 2 ( mod p) , … , (p-1)a ≡ r p-1 ( mod p)         Since r 1 , r 2 , … r p-1 are remainders obtained when a,2a,…,(p-1)a are divided by p.         ∴ r 1. r 2 . … .r p-1 = 1 . 2 .3 …. . (p-1).         Multiplying the above congruent relations : a . 2a. … . (p-1) ≡ r 1 r 2 …r p-1 ( mod p ) ð   {1.2…....

                                                                                                               Congruences :                  The property of congruence provides a way of classifying integers according to the remainder      obtained  upon division by a fixed positive integer. In fact the remainder is the only thing of  interest.  In this section we study a relation on the integers that is defined in terms of remainders. Definition :                    Let m be a fixed positive integer and a,b 𝛜 Z. 'a'  is said to be " congruent to          ...

                           E. T. Bell used formal power series to study properties of multiplicative arithmetical functions . Definition : [ Bell Series ]    Given an arithmetical function f and a prime p , we denote by f p (x) the formal power series         f p (x) = 𝜮 f(p n ) x n and call this the Bell series of f modulo p. Note :           Bell series are especially useful when f is multiplicative. Examples : ·          The Bell series for the Mobius function 𝛍 is given by 𝛍 p (x) = 1-x. ·          The Bell series for the Euler ‘s totient function ⲫ is given by                 ⲫ p (x)  =   (1-x) / (1-px) ·          The Bell series ...

Formal power series: In calculus an infinite series of the form 𝜮 a(n)  = a(0) + a(1) x + a(2) x² +...+ a(n) x +... is called a power series in x. Here both x and a(n) are real or complex numbers. To each power series there corresponds a radius of convergence r > 0 such that the series converges absolutely if | x |<r and diverges if | x | > r. Note:          Here the radius r can be +∞ Here in this, we consider power series from a different point of view. We call these power series as FORMAL power series to distinguish them from the ordinary power series of calculus. In the formal power series, x is never assigned a numerical value. In power series 𝜮 c(n) x", the symbol x" is simply a device for locating the position of the nth coefficient a(n). The coefficient a(0) is called the constant coefficient of the series. Let A(x)=a(n) x"; B(x) = b(n) x. Then 1. A(x)+B(x) iff a(n) = b(n) for all n > 0 2. A(x)+B(x)=(a(x)+b(x)) x". 3. A(x) B(x) = c(n) ...