An algebra problem by Shriram Lokhande

Algebra Level 3

Define $\varrho(n)$ be a function which gives product of roots of the equation $\sqrt{n}^x+\frac{1}{\sqrt{n}^x}=2$ where $n$ is a gaussian integer .

Find $\prod_{gaussian integers}{\varrho(n)}$

Details

gaussian integer is an complex number $a+bi$ where a and b as integers

The answer is 0.

1 solution

Shriram Lokhande
Jul 24, 2014

As 2 is an gaussian integer & $\sqrt{2}^0=1$ we get $\varrho(2)=0$ ,

which makes whole product $\displaystyle \prod_{gaussian integer}\varrho(n)=0$

I am very eager to know(& working) whether there are any values other than $n=1$ & $x=0$ which will satisfy the equation .

$\sqrt{n}^x+\frac{1}{\sqrt{n}^x}=2\\ \implies n^x-2\sqrt{n}^x+1=0\\ \implies (\sqrt{n}^x-1)^2=0$

Now, when a variable $p$ varies over all complex numbers, the equation $p^2=0$ has only one solution which is $p=0$ . Similarly, we have,

$\sqrt{n}^x-1=0\implies \sqrt{n}^x=1\\ \implies n^x=1$

From this result, we can conclude that $x=0$ always satisfies the given equation $\forall n\in \mathbb{C}$ , where $\mathbb{C}$ represents the set of complex numbers.

Also, we know that the set of Gaussian integers $\mathbb{Z[i]}\subset \mathbb{C}$ .

So, the product turns out to be $\boxed{0}$

P.S - Finding one example that there is the element $0$ in the product is enough though. This comment answers what you were eager to know. :D

Hope this helps... @Shriram Lokhande

Prasun Biswas - 6 years, 3 months ago

An algebra problem by Shriram Lokhande

The answer is 0.

1 solution

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