Question-69064

Question Number 69064 by Sayantan chakraborty last updated on 18/Sep/19

Commented by Sayantan chakraborty last updated on 18/Sep/19

PLEASE HELP

Commented by Sayantan chakraborty last updated on 19/Sep/19

please solve it .

Commented by MJS last updated on 20/Sep/19

y = {(7 + 3 \sqrt{6})}^{n}

y {y} \notin Z \forall n = 2 k; k \in N^{★} [? [

y {y} = 5^{n} \in Z \forall n = 2 k - 1; k \in N^{★} [?]

\Rightarrow

f (k) = 5^{2 k - 1} + 25^{2 k - 1} + 125^{2 k - 1}

x = f (45) = 5^{89} + 25^{89} + 125^{89}

we have to show for which values of k

31 ∣ (5^{2 k - 1} + 25^{2 k - 1} + 125^{2 k - 1}) [?]

trying \dots

{it}^{'} s not true for k \in {2, 5, 8, 11, 14, 17, 20, \dots}

looks like {it}^{'} s not true for k = 2 + 3 l; l \in N [?]

n = 89 \Leftrightarrow k = 45 \Leftrightarrow k \neq 2 + 3 l \forall l \in N

\Rightarrow 31 ∣ (x {x} + x^{2} {x}^{2} + x^{3} {x}^{3}); x = {(7 + 3 \sqrt{6})}^{89}

needed proofs are marked with [?]

Answered by mind is power last updated on 19/Sep/19

i w i l l t r y i s n o t e a s y!

Commented by Sayantan chakraborty last updated on 20/Sep/19

It is challenging problem