Prove that any number of the form pq, where p and q are prime numbers greater than 2, can be written as the difference of two squares in exactly two distinct ways.

If we want to prove it, we need to prove every odd number can be expressed as the difference of two squares, which is very easy.

Suppose this odd number to be 2n-1, then we can see 2n-1=n2-(n-1)2

Then we let pq=a2-b2=(a-b)(a+b).Then we can see either p=a-b & q=a+b or 1=a-b & pq=a+b, which are two different forms of squares.

SL
Answered by Shibo L. STEP tutor

7264 Views

See similar STEP University tutors

Related STEP University answers

All answers ▸

Suppose that 3=2/x(1)=x(1)+(2/x(2))=x(2)+(2/x(3))=x(3)+(2/x(4))+...Guess an expression, in terms of n, for x(n). Then, by induction or otherwise, prove the correctness of your guess.


Evaluate the integral \int \frac{x}{x tan(x) + 1} dx using integration by substitution, hence evaluate \int \frac{x}{x cot(x) - 1} dx.


Show that i^i = e^(-pi/2).


By use of calculus, show that x − ln(1 + x) is positive for all positive x.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences