OCR MEI
Further Pure with Technology
2019
June
— Question 2
Exam Board
OCR MEI
Module
Further Pure with Technology (Further Pure with Technology)
Year
2019
Session
June
Topic
Number Theory
2
Prove that if \(x\) and \(y\) are integers which satisfy \(x ^ { 2 } - 2 y ^ { 2 } = 1\), then \(x\) is odd and \(y\) is even.
Create a program to find, for a fixed positive integer \(s\), all the positive integer solutions \(( x , y )\) to the equation \(x ^ { 2 } - 2 y ^ { 2 } = 1\) where \(x \leqslant s\) and \(y \leqslant s\). Write out your program in the Printed Answer Booklet.
Use your program to find all the positive integer solutions \(( x , y )\) to the equation \(x ^ { 2 } - 2 y ^ { 2 } = 1\) where \(x \leqslant 600\) and \(y \leqslant 600\). Give the solutions in ascending order of the value of \(x\).
By writing the equation \(x ^ { 2 } - 2 y ^ { 2 } = 1\) in the form \(( x + \sqrt { 2 } y ) ( x - \sqrt { 2 } y ) = 1\) show how the first solution (the one with the lowest value of \(x\) ) in your answer to part (c) can be used to generate the other solutions you found in part (c).
What can you deduce about the number of positive integer solutions \(( x , y )\) to the equation \(x ^ { 2 } - 2 y ^ { 2 } = 1\) ?
In the remainder of this question \(T _ { m }\) is the \(m ^ { \text {th } }\) triangular number, the sum of the first \(m\) positive integers, so that \(T _ { m } = \frac { m ( m + 1 ) } { 2 }\).
Create a program to find, for a fixed positive integer \(t\), all pairs of positive integers \(m\) and \(n\) which satisfy \(T _ { m } = n ^ { 2 }\) where \(m \leqslant t\) and \(n \leqslant t\). Write out your program in the Printed Answer Booklet.
Use your program to find all pairs of positive integers \(m\) and \(n\) which satisfy \(T _ { m } = n ^ { 2 }\) where \(m \leqslant 300\) and \(n \leqslant 300\). Give the pairs in ascending order of the value of \(m\).
By comparing your answers to part (c) and part (g), or otherwise, prove that there are infinitely many triangular numbers which are perfect squares.