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hihihi · Jul 15, 2016

Find integers a and b such that:
(a) (1 +√3 )^5 = a + b√3

InteGrand · Jul 15, 2016

hihihi said:
Find integers a and b such that:
(a) (1 +√3 )^5 = a + b√3

$\noindent Expanding the LHS using $\left(1+x\right)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5$ with $x=\sqrt{3}$, it is $1 + 5 \sqrt{3} + 10\times 3 + 10 \times 3\sqrt{3} + 5 \times 3^2 + 1 \times 3^2 \sqrt{3} = 1 + 5\sqrt{3} + 30 + 30\sqrt{3} + 45 + 9\sqrt{3} = 76 + 44\sqrt{3}$. So the integers $a=76,b=44$ work (and in fact is the only solution).$

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hihihi

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