find-all-solutions-if-exist-of-x-2-5y-2-2016-with-x-y-N-

Question Number 75660 by mr W last updated on 15/Dec/19

f i n d a l l s o l u t i o n s (i f e x i s t) o f

x^{2} + 5 y^{2} = 2016

w i t h x, y \in N .

Commented by mathmax by abdo last updated on 15/Dec/19

5 i s p r i m e l e t c o n s i d e r c o n g r u e n c e m o d u l o 5

(e) \Rightarrow {(\bar{x})}^{2} + 0 = 201 \bar{6} = 1 \bar{6} = \bar{1} \Rightarrow \overset{-^{2}}{x} - \bar{1} = \bar{0} \Rightarrow

(\bar{x} - \bar{1}) (x + \bar{1}) = \bar{0} Z / 5 Z i s a c o r p s \Rightarrow \bar{x} - \bar{1} = 0 o r \bar{x} + 1 = 0 \Rightarrow

x = 5 k + 1 o r x = 5 k - 1 (k \in Z) n o w r e s t t o f i n d t h e v a l u e s o f k

w i c h v e r i f y t h e e q u a t i o n w i t h x^{2} ⩽ 2016 a n d y^{2} ⩽ [\frac{2016}{5}] \dots

Answered by vishalbhardwaj last updated on 15/Dec/19

x^{2} = 2016 - {5 y}^{2} ⩾ 0

\Rightarrow {5 y}^{2} ⩽ 2016

\Rightarrow y^{2} ⩽ 403.20

\Rightarrow y ⩽ \sqrt{403.20} \Rightarrow 1 ⩽ y ⩽ 20, y \in N

similarily {5 y}^{2} = 2016 - x^{2}

\Rightarrow y^{2} = \frac{2016 - x^{2}}{5} ⩾ 0

\Rightarrow 2016 - x^{2} ⩾ 0

\Rightarrow x^{2} ⩽ 2016 \Rightarrow 1 ⩽ x ⩽ 44, x \in N

when x = 4 then y = 20

and when x = 36 then y = 12

\Rightarrow we have two solutions : (x = 4, y = 20)

and (x = 36, y = 12)

Commented by mr W last updated on 15/Dec/19

t h a n k s a l o t s i r .

Commented by MJS last updated on 15/Dec/19

one is missing : x = 44, y = 4

Answered by mind is power last updated on 15/Dec/19

x^{2} + {5 y}^{2} = 2016

x^{2} + {5 y}^{2} = 0 (4)

\Leftrightarrow x^{2} + y^{2} = 0 (4)

x^{2} = 0, 1 (4) \Rightarrow x^{2} = y^{2} \equiv 0 (4)

\Leftrightarrow x = 2 k y = 2 t

\Rightarrow 4 (k^{2} + {5 t}^{2}) = 2016 \Rightarrow k^{2} + {5 t}^{2} = 504 \equiv 0 (4) like previous congrunce \Rightarrow

\Rightarrow k = 2 s, t = 2 z \Rightarrow t^{2} + {5 z}^{2} = 126 if z \equiv 0 (2) \Rightarrow t \equiv 0 (2)

\Rightarrow t^{2} + {5 z}^{2} = 126 \equiv 0 (4) absurde \Rightarrow z = 2 y + 1

t^{2} = 126 - {5 z}^{2} \Rightarrow z ⩽ \sqrt{\frac{126}{5}} \Rightarrow z ⩽ 5 \Rightarrow z \in {1, 3, 5}

z = 1 \Rightarrow t^{2} = 121 \Rightarrow t = 11

z = 3 \Rightarrow t^{2} = 81 \Rightarrow t = 9

z = 5 \Rightarrow t^{2} = 1 \Rightarrow t = 1

z = 1 \Rightarrow t = 2 \Rightarrow y = 4

t = 11 \Rightarrow x = 44

z = 5 \Rightarrow y = 20

t = 1 \Rightarrow x = 4

z = 3 \Rightarrow y = 12 \Rightarrow x = 36

S = {(4, 20); (44, 4); (36, 12)}

Commented by mr W last updated on 15/Dec/19

t h a n k s a l o t s i r!

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