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# 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
9 months, 3 weeks 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. · 9 months, 3 weeks ago

What about $$n = 1$$? Staff · 9 months, 3 weeks ago

Ahahaha, you've found a loophole. Let me fix it. · 9 months, 3 weeks 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 · 9 months, 3 weeks 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. · 9 months, 3 weeks 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 · 9 months, 3 weeks ago

HA! That's a nice interpretation. ThankYou · 9 months, 3 weeks ago

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

Approach 2: Use Quadratic Diophantine Equations - Pell's Equation. Staff · 9 months, 3 weeks ago