how-many-natural-solution-are-there-for-x-2-y-2019-

Question Number 84047 by jagoll last updated on 09/Mar/20

how many

natural solution are there for

x^{2} - y! = 2019 .

Answered by naka3546 last updated on 09/Mar/20

1, s i r n a m e l y (x, y) = (45, 3)

Commented by jagoll last updated on 09/Mar/20

yes . 2019 + 3! = 2025

2019 + 6 = 45^{2}

Answered by mind is power last updated on 10/Mar/20

\Rightarrow y! = x^{2} - 2019 \Rightarrow x ⩾ 45

2019 = 3.673

c a s e 1, y ⩾ 3

i f y ⩾ 3 \Rightarrow x^{2} = 2019 + y!

3 ∣ y! s i n c e y ⩾ 3

x^{2} = 3.673 + y!

\Rightarrow 3 ∣ x^{2} s i n c e 3 i s p r i m \Rightarrow 3 ∣ x

\Rightarrow 9 ∣ x^{2}

\Rightarrow x^{2} = 2019 + y! \equiv 0 [9]

\Rightarrow y! \equiv - 2019 [9] \equiv - 2016 - 3 [9] \equiv 6 [9]

\Rightarrow y ⩽ 5

y = 5 \Rightarrow y! \equiv 3 (9)

y = 4 \Rightarrow y! \equiv 6 (9)

y = 3 \Rightarrow y! \equiv 6 (9)

\Rightarrow y \in {3, 4}

y = 4 \Rightarrow x^{2} = 2019 + 4! = 2043 n o t p o s s i b l

y = 3 \Rightarrow x^{2} = 2019 + 6 = 2025 \Rightarrow x = \sqrt{2025} = 45 g o t o n e

i f y ⩽ 2

y = 0 o r 1 \Rightarrow x^{2} = 2019 + 1 = 2020 n o t p o s s i b l e

y = 2 \Rightarrow x^{2} = 2019 + 2 \Rightarrow x = 2021

w e c a n u s e x ⩾ 45 \Rightarrow x^{2} ⩾ 2025

y! + 2019 ⩾ 2025 \Rightarrow x^{2} ⩾ 2025 - 2019 = 6 \Rightarrow y ⩾ 3

s o w e u s e j u s t c a s e o n e

\Rightarrow (x, y) \in {(45, 3)}