--%>
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 : 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 : State Fermat algorithm The basic Fermat

    The basic Fermat algorithm is as follows: Assume that n is an odd positive integer. Set c = [√n] (`ceiling of √n '). Then we consider in turn the numbers c2 - n; (c+1)2 - n; (c+2)2 - n..... until a perfect square is found. If th

    		•
  • Q : Calculus I need it within 4 hours. Due

    I need it within 4 hours. Due time March 15, 2014. 3PM Pacific Time. (Los Angeles, CA)

    		•
  • Q : Properties for polynomial Specify the

    Specify the important properties for the polynomial.

    		•
  • Q : Explain the work and model proposed by

    Explain the work and model proposed by Richardson.

    		•
  • Q : Problem on Nash equilibrium In a

    In a project, employee and boss are working altogether. The employee can be sincere or insincere, and the Boss can either reward or penalize. The employee gets no benefit for being sincere but gets utility for being insincere (30), for getting rewarded (10) and for be

    		•
  • Q : Set Theory & Model of a Boolean Algebra

    II. Prove that Set Theory is a Model of a Boolean Algebra The three Boolean operations of Set Theory are the three set operations of union (U), intersection (upside down U), and complement ~.  Addition is set

    		•
  • Q : Logic and math The homework is attached

    The homework is attached in the first two files, it's is related to Sider's book, which is "Logic for philosophy" I attached this book too, it's the third file.

    		•
  • Q : Uniform scaling what is uniform scaling

    what is uniform scaling in computer graphic

    		•
  • 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.

    		•