# Try to solve this

Prove that always exist integer number a,b,c such as $0< \left | a+b\sqrt{2} +c\sqrt{3}\right |< \frac{1}{1000}$

Note by Ms Ht
2 years, 8 months ago

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Hint: Show that for positive even integer $$n$$, $$(\sqrt 3 + \sqrt 2)^n + (\sqrt3 - \sqrt2)^n$$ is always an integer.

- 2 years, 8 months ago

What about $$n = 1$$?

Staff - 2 years, 8 months ago

Ahahaha, you've found a loophole. Let me fix it.

- 2 years, 8 months ago

Nope, not fixed yet. It's not true for odd integers.

Essentially what you are doing is my approach 1 but with setting $$a = 0$$ (which complicates it just slightly).

Staff - 2 years, 8 months ago

Awhhh!! I should have known! Fixed again~

Hmm... I don't see how mine is slightly harder than Approach 1, let me put my thinking cap on.

- 2 years, 8 months ago

The slightly harder part being why we have to use $$2n$$ instead.

$$( 1 + \sqrt{2} ) ^n + ( 1 - \sqrt{2} ) ^ n$$ is an integer for all $$n$$, because odd powers of 1 is still an integer. Applying your approach results in $$c = 0$$.

Staff - 2 years, 8 months ago

HA! That's a nice interpretation. ThankYou

- 2 years, 8 months ago

Approach 1: Simplify the problem by setting $$c = 0$$.

Approach 2: Use Quadratic Diophantine Equations - Pell's Equation.

Staff - 2 years, 8 months ago