Algebra

**Zero266** · November 17th 2009, 03:42 PM

if a and b are integers and (a+b* sqrt(11) )> 1 and (a+b* sqrt(11) )* (a-b* sqrt(11) )=1, how do i show that a and b are positive integers? I was able to prove b>0 , but i can't seem to get the a >0 part, which makes me doubt whether i really showed b>0 .

Thanks

**Sampras** · November 17th 2009, 06:03 PM

Suppose $a+b\sqrt{11} > 1$ . Also, $(a+b\sqrt{11})(a-b\sqrt{11}) = 1$ . Let $c := a+b \sqrt{11}$ . Then $a-b \sqrt{11} = \frac{1}{c}$ . Then $ca-cb \sqrt{11} = 1$ . Or $\sqrt{11} = - \frac{\frac{1}{c}-a}{b}$ . Or $\frac{1}{\sqrt{11}} = -\frac{b}{\frac{1}{c}-a}$ .

We know that $0 < \frac{1}{c} < 1$ . So $a > 0$ . Also $b > 0$ , which forces the quotient to be positive. If $b < 0$ then $a < 0 \implies a+b \sqrt{11} < 1$ . So $a >0$ so that the quotient is negative.

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