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#### CaptainBlack

##### Well-known member

- Jan 26, 2012

- 890

(a) Recall why there are no integer solutions \(m, n \in \mathbb{N}\) to the equation \(m^2 = 2n^2\).

ANSWER = an irrational number cannot be expressed as a fraction

(b) Show that if \(m, n \in \mathbb{N}\) are integer solutions to the equation

\(m^2 = 2n^2 + 1\), (*)

then so are \(M = m^2 + 2n^2\) and \(N = 2mn\), namely \(M^2 = 2N^2 + 1\).

ANSWER = sub them in and then they equal each other

(c)Give a simple solution \(m, n \in \mathbb{N}\) to equation (*).

ANSWER = When \(m=3\) and \(n=2\)

(d )Deduce that there are infinitely many pairs of integers \(m, n \in \mathbb{N}\) satisfying (*).

ANSWER = Help????? A hint is that \(M^2 = 2N^2 + 1\) is significantly larger than (*)

(e) Let \(m, n \in \mathbb{N}\) be any pair of integers satisfying equation (*).

Show that if \(p \in \mathbb{N}\) is a prime number then

\(p | n\) implies that \p doesn’t divide m.

[Use the fact that if \(p\) is prime and \(a, b \in \mathbb{N}\), and \(p | ab\) then \(p | a\) or \( p | b\)].

ANSWER = I done this so it’s ok but have included it anyway

( f ) What does it mean for a fraction \(a/b\) to be in reduced form? ANSWER = done

Explain why if \(m, n \in \mathbb{N}\) satisfy equation (*), then is \(m/n\) in reduced form. ANSWER = Help??????

(g) Use the above to generate a fraction which approximates \(\sqrt{2}\) to 5 decimal places.

ANSWER = Help???????? This is the main bit in which I require help as you have to use the question to generate the approximation.

So mainly g, a bit of f and d is what I need help with thanks!