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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 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.

Pi Han Goh - 2 years ago

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What about \( n = 1 \)?

Calvin Lin Staff - 2 years ago

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

Pi Han Goh - 2 years ago

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@Pi Han Goh 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).

Calvin Lin Staff - 2 years ago

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@Calvin Lin 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.

Pi Han Goh - 2 years ago

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@Pi Han Goh 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 \( c = 0 \).

Calvin Lin Staff - 2 years ago

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@Calvin Lin HA! That's a nice interpretation. ThankYou

Pi Han Goh - 2 years ago

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Approach 1: Simplify the problem by setting \( c = 0 \).

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

Calvin Lin Staff - 2 years ago

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