If-x-2-y-2-2023-2024-amp-x-y-N-how-many-set-x-y-

Question Number 202677 by BaliramKumar last updated on 31/Dec/23

If x^{2} - y^{2} = 2023^{2024} & x, y \in N

how many set {x, y}

Answered by AST last updated on 31/Dec/23

x^{2} - y^{2} \equiv 7^{2024} \times 17^{4048} = (x - y) (x + y)

\Rightarrow x - y = 7^{a} 17^{b} \land x + y = 7^{2024 - a} 17^{4048 - b}

\Rightarrow (x, y) = (\frac{7^{2024 - a} 17^{4048 - b} + 7^{a} 17^{b}}{2}, \frac{7^{2024 - a} 17^{4048 - b} - 7^{a} 17^{b}}{2})

s u c h t h a t x ⩾ y \Rightarrow a \in {0, 1, 2, \dots, 2024};

b \in {0, 1, 2, \dots, 4048} \Rightarrow T h e r e a r e \frac{4049 \times 2025 - 1}{2}

Commented by BaliramKumar last updated on 31/Dec/23

set {x, y} = s e t {y, x} a n d x a n d y n o n z e r o

If a = 1012 & b = 2024 then y = 0

Commented by AST last updated on 31/Dec/23

{W h a t}^{'} s t h e a n s w e r ?

Commented by BaliramKumar last updated on 31/Dec/23

⌊ \frac{(2024 + 1) (4048 + 1)}{2} ⌋ = 4099612

Any computer programmer please

check the answer

Commented by MathematicalUser2357 last updated on 06/Jan/24

Right

Answered by witcher3 last updated on 31/Dec/23

2023^{2024} = 17^{4048} . 7^{2024}

(x + y) (x - y) = 2023^{2024}

{\begin{cases} x + y = 17^{a} . 7^{b} \\ x - y = 17^{4048 - a} 7^{2024 - b} \end{cases}

solve in Z

\Rightarrow 2 x = 17^{a} . 7^{b} + 17^{4048 - a} 7^{2024 - b}

existe solution if and only if

(a, b) \neq {(0, 0); (4048, 2024)}

we have all set of type {(x, y); (x - y)}, (x, y) \in N^{2}

a \in [0, 4048], b \in [0, 2024]

B = card {x, - y} = card {x, y} = A

A \cap B = {2023^{1012}, 0}

card (A \cup B) = 2 card (A) - 1

2 . card {x, y} - 1 = (4049) . (2025) - 2

card {x, y} = \frac{(4049.2025 - 1)}{2} = 4099612

Answered by BaliramKumar last updated on 01/Jan/24

2023^{2024} = {(7 \times 17^{2})}^{2024} = 7^{2024} \times 17^{4048}

No . of set {x, y} = ⌊ ∣ \frac{No . of even factors - No . of odd factors}{2} ∣ ⌋

No . of set {x, y} = ⌊ ∣ \frac{0 - (2024 + 1) (4048 + 1)}{2} ∣ ⌋

No . of set {x, y} = ⌊ ∣ \frac{- (2025) (4049)}{2} ∣ ⌋

No . of set {x, y} = ⌊ \frac{8199225}{2} ⌋

No . of set {x, y} = ⌊ 4099612.5 ⌋

No . of set {x, y} = 4099612