--%>
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 : 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 : 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 : Area Functions & Theorem Area Functions

    Area Functions 1. (a) Draw the line y = 2t + 1 and use geometry to find the area under this line, above the t - axis, and between the vertical lines t = 1 and t = 3. (b) If x > 1, let A(x) be the area of the region that lies under the line y = 2t + 1 between t

    		•
  • Q : Row-echelon matrix Determine into which

    Determine into which of the following 3 kinds (A), (B) and (C) the matrices (a) to (e) beneath can be categorized:       Type (A): The matrix is in both reduced row-echelon form and row-echelon form. Type (B): The matrix

    		•
  • Q : Theorem-G satis es the right and left

    Let G be a group. (i) G satis es the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b. (ii) If g ≡ G, then (g-1)

    		•
  • Q : Law of iterated expectations for

     Prove the law of iterated expectations for continuous random variables. 2. Prove that the bounds in Chebyshev's theorem cannot be improved upon. I.e., provide a distribution that satisfies the bounds exactly for k ≥1, show that it satisfies the bounds exactly, and draw its PDF. T

    		•
  • 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:     

    		•
  • Q : Elementary Logic Set & Model of a

    Prove that Elementary Logic Set is a Model of a Boolean Algebra The three Boolean operations of Logic are the three logical operations of  OR ( V ), AN

    		•
  • Q : Use MS Excel to do the computations

    Select a dataset of your interest (preferably related to your company/job), containing one variable and atleast 100 data points. [Example: Annual profit figures of 100 companies for the last financial year]. Once you select the data, you should compute 4-5 summary sta

    		•
  • 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

    		•