1-Find-all-natural-pairs-of-integers-x-y-such-that-x-3-y-3-xy-61-2-Find-gcd-x-4-x-3-x-3-x-

Question Number 104100 by bramlex last updated on 19/Jul/20

(1) F i n d a l l n a t u r a l p a i r s o f

i n t e g e r s (x, y) s u c h t h a t x^{3} - y^{3} = x y + 61 .

(2) F i n d g c d (x^{4} - x^{3}, x^{3} - x)

Answered by bemath last updated on 19/Jul/20

(2) x^{4} - x^{3} = x^{3} (x - 1) = x^{2} . x (x - 1)

x^{3} - x = x (x^{2} - 1) = x (x - 1) (x + 1)

s o g c d (x^{4} - x^{3}, x^{3} - x) =

x (x - 1) o r x^{2} - x ★

Answered by john santu last updated on 19/Jul/20

(1) x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

\Leftrightarrow (x - y) (x^{2} + x y + y^{2}) = x y + 61

n o t i c e t h a t x > y . t h e r e f o r e w e

h a v e t o c o n s i d e r x^{2} + x y + y^{2} ⩽

x y + 61 o r x^{2} + y^{2} ⩽ 61 .

b e c a u s e x > y, w e h a v e

61 ⩾ x^{2} + y^{2} ⩾ 2 y^{2} \Rightarrow y \in {1, 2, 3, 4, 5}

{\begin{cases} y = 1; x^{3} - x - 62 = 0 \\ y = 2; x^{3} - 2 x - 69 = 0 \\ y = 3; x^{3} - 3 x - 89 = 0 \\ y = 4; x^{3} - 4 x - 125 = 0 \\ y = 5; x^{3} - 5 x - 186 = 0; x = 6 \end{cases}

w e s e e t h e o n l y w o r k i n g v a l u e

f o r x i s w h e n x = 6, y = 5; s o t h e

o n l y n a t u r a l p a i r s o f s o l u t i o n

i s (x, y) = (6, 5) (J S ⊛)

Commented by bramlex last updated on 19/Jul/20

t h a n k y o u

Answered by floor(10²Eta[1]) last updated on 19/Jul/20

(2) \gcd (x^{4} - x^{3}, x^{3} - x) = \gcd (x^{3} (x - 1), (x - 1) x (x + 1))

= x (x - 1) . \gcd (x^{2}, x + 1) ★

d = \gcd (x^{2}, x + 1) \Rightarrow d ∣ x^{2} \land d ∣ x + 1

\Rightarrow d ∣ (- 1) . x^{2} + x (x + 1) = x

\Rightarrow \gcd (x^{2}, x + 1) = \gcd (x, x + 1) = 1

\gcd (x^{4} - x^{3}, x^{3} - x) =

★ x (x - 1)