what-are-the-solutions-of-3x-2-1-n-where-n-N-

Question Number 73308 by wo1lxjwjdb last updated on 10/Nov/19

w h a t a r e t h e s o l u t i o n s

o f \sqrt{3 x^{2} + 1} = n w h e r e n \in N

Commented by MJS last updated on 10/Nov/19

n_{k} = \frac{1}{2} ({(2 - \sqrt{3})}^{k} + {(2 + \sqrt{3})}^{k}) = \cosh (k \ln (2 - \sqrt{3}))

x_{k} = - \frac{\sqrt{3}}{3} \sinh (k \ln (2 - \sqrt{3}))

k, n_{k}, x_{k} \in N

Commented by MJS last updated on 10/Nov/19

you mean n \in N

Commented by kaivan.ahmadi last updated on 10/Nov/19

n \in N

3 x^{2} + 1 = n^{2} \Rightarrow 3 x^{2} = n^{2} - 1 \Rightarrow x^{2} = \frac{n^{2} - 1}{3} \Rightarrow

x = \sqrt{\frac{n^{2} - 1}{3}}

Commented by wo1lxjwjdb last updated on 10/Nov/19

w h e n i s i t a w h o l e n u m b e r ?

Commented by MJS last updated on 10/Nov/19

n \in N

x \in ?

if x \in R

x = \pm \sqrt{\frac{n^{2} - 1}{3}}; n > 0

[for x \in C we also get n = 0; x = \frac{\sqrt{3}}{3} i]

if x \in N

same equation, solutions are :

x \in {0, 1, 4, 15, 56, 209, 780, \dots}

n \in {1, 2, 7, 26, 97, 362, 1351, \dots}

Commented by kaivan.ahmadi last updated on 10/Nov/19

M r M J S a n s w e r e d t h i s q u o e s t i o n .