--%>
Assignment Help - homework help

Explain Factorisation by Fermats method

Factorisation by Fermat's method: This method, dating from 1643, depends on a simple and standard algebraic identity. Fermat's observation is that if we wish to nd two factors of n, it is enough if we can express n as the di fference of two squares. This is because if n = a2 - b2, then we have immediately

n = a2 - b2 = (a+b)(a - b);

and so we have found two factors, a+b and a - b, of n.

It is possible here that a - b might equal 1, in which case we will only have found the trivial factorisation n = n x 1, but we can arrange matters so that this will only happen if n has no other factorisation - i.e., is prime.

At first glance, it may seem over-optimistic to hope that an expression for n as the di fference of two squares will exist.

But assume that n is odd, which we can always do if we are trying to factorise n. Then if n = uv and we put

a = 1/2(u+v) and b = 1/2(u - v);

we have n = a2 - b2 (note that a and b are both integers if n is odd), so that a representation of n as the difference of two squares does exist. (In fact, it is easy to see that the above formulae define a one-to-one correspondence between representations of n as the di erence of two squares and as the product of two factors - exercise.)

   Related Questions in Mathematics

  • Q : Uniform scaling what is uniform scaling

    what is uniform scaling in computer graphic

    		•
  • Q : Relationships Between Data Introduction

    Relationships Between Data - Introduction to Linear Regression Simple Regression Notes If you need guidance in terms of using Excel to run regressions, check pages 1 - 10 of the Excel - Linear Regression Tutorial posted to th

    		•
  • Q : Problem on inverse demand curves In

    In differentiated-goods duopoly business, with inverse demand curves: P1 = 10 – 5Q1 – 2Q2P2 = 10 – 5Q2 – 2Q1 and per unit costs for each and every firm equal to 1.<

    		•
  • Q : Numerical solution of PDE this

    this assignment contains two parts theoretical and coding the code has to be a new. old code and modified code will appear in the university website .

    		•
  • Q : Explain Factorisation by Fermats method

    Factorisation by Fermat's method: This method, dating from 1643, depends on a simple and standard algebraic identity. Fermat's observation is that if we wish to nd two factors of n, it is enough if we can express n as the di fference of two squares.

    		•
  • Q : First-order formulas over the

    Consider the unary relational symbols P and L, and the binary relational symbol On, where P(a) and I(a) encode that a is apoint and a (sraight) line in the 2-dimensional space, respectively, while On(a,b) encodes  that a is a point, b is a line, and o lies on b.

    		•
  • Q : Explain Factorisation by trial division

    Factorisation by trial division: The essential idea of factorisation by trial division is straightforward. Let n be a positive integer. We know that n is either prime or has a prime divisor less than or equal to √n. Therefore, if we divide n in

    		•
  • Q : Numerical solution of PDE i want you to

    i want you to solve this assignment. this consist of two parts theoretical and coding. the code has to be created by you. no modified or copying code. you have to mention the exact solution and the proportion error. also you have to explain the sketch that you get from the code. these information

    		•
  • Q : Formal logic It's a problem set, they

    It's a problem set, they are attached. it's related to Sider's book which is "Logic to philosophy" I attached the book too. I need it on feb22 but feb23 still work

    		•
  • Q : Problem on budgeted cash collections

    XYZ Company collects 20% of a month's sales in the month of sale, 70% in the month following sale, and 5% in the second month following sale. The remainder is not collectible. Budgeted sales for the subsequent four months are:     

    		•