Suppose $a,b$ are coprime and $0 Suppose $m_1a^4+m_2a^3b+m_3a^2b^2+m_4ab^3+m_5b^4=0$ where $m_i\in\Bbb Z$ holds and no $m_i=0$, is there a way to show $|m_i|>a^2$ at least for one $i$ holds?
On minimum coefficients.
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elementary-number-theory
1 Answers
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NO. Let $a=2$ and $b=3.$ Then $$3a^4+a^3b-2a^2b^2+3ab^3-2b^4=$$ $$= 3(16)+24-2(36)+3(54)-2(81)=$$ $$=48+24-72+162-162=0.$$
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0My feeling was that it was false so I tried the smallest possible $a,b$ and,noting that this required $m_5$ to be even, I played with some values until I got it. It often helps to look at particular cases. If that doesn't solve it, it may give some insight. – 2017-02-13