Find-the-positive-integer-solution-of-the-equation-x-3-y-3-7-14xy-3-

Question Number 157163 by MathSh last updated on 20/Oct/21

Find the positive integer solution

of the equation :

x^{3} + y^{3} = 7 \cdot (14 xy + 3)

Commented by Rasheed.Sindhi last updated on 22/Oct/21

x^{3} + y^{3} = 7 \cdot (14 xy + 3)

x^{3} + y^{3} - 21 = 7^{2} . 2 xy

x^{3} + y^{3} - 21 \in E

\Rightarrow x^{3} + y^{3} \in O

\Rightarrow one of x, y is odd and other is even .

As x and y are exchangeable we may

take any one of these as odd .

let x \in O, y \in E

{(2 m + 1)}^{3} + {(2 n)}^{3} - 21 = 98 (2 m + 1) (2 n)

▸ {8 m}^{3} + 1 + 6 m (2 m + 1) + {8 n}^{3} - 21

= 98 (2 m + 1) (2 n)

▸ {8 m}^{3} + {8 n}^{3} + 6 m (2 m + 1) - 20

= 98 (2 m + 1) (2 n)

▸ {4 m}^{3} + {4 n}^{3} + 3 m (2 m + 1) - 10

= 49 (2 m + 1) (2 n)

49 (2 m + 1) (2 n) \in E

\Rightarrow {4 m}^{3} + {4 n}^{3} + 3 m (2 m + 1) - 10 \in E

\Rightarrow 3 m (2 m + 1) \in E

\Rightarrow {6 m}^{2} + 3 m \in E \Rightarrow m \in E

x = 2 m + 1 = 2 (2 k) + 1 = 4 k + 1; k \in W

x = 1, 5, 9, \dots .

x = 1 :

{(1)}^{3} + y^{3} = 7 (14.1 . y + 3)

y^{3} - 98 y - 20 = 0

(y - 10) (\underset{y \notin N}{\underset{⏟}{y^{2} + 10 y + 2}}) = 0

y = 10

Continue

Answered by Rasheed.Sindhi last updated on 20/Oct/21

x^{3} + y^{3} = 7 \cdot (14 xy + 3); \overset{?}{x}, \overset{?}{y} \in Z^{+}

{(\underset{u}{\underset{⏟}{x + y}})}^{3} - 3 \underset{v}{\underset{⏟}{xy}} (x + y) = 7 \cdot (14 xy + 3)

u^{3} - 3 uv = 7 (14 v + 3)

u (u^{2} - 3 v) = 7 (14 v + 3)

case 1 : u = 7 \land u^{2} - 3 v = 14 v + 3

49 - 3 v - 14 v - 3 = 0

17 v = 46 \Rightarrow v \notin Z^{+} \dots \dots \dots (i)

case 2 : u^{2} - 3 v = 7 \land u = 14 v + 3

{(14 v + 3)}^{2} - 3 v = 7

{196 v}^{2} + 84 v + 9 - 3 v - 7 = 0

{196 v}^{2} + 81 v + 2 = 0

v \notin Z^{+} \dots \dots \dots \dots \dots \dots \dots . (ii)

In both possible cases :

v \notin Z^{+} \Rightarrow xy \notin Z^{+} \Rightarrow x \notin Z^{+} \lor y \notin Z^{+}

Solution not possible!

Commented by MathSh last updated on 21/Oct/21

Thank you dear Ser, but ans : (1; 10) (10; 1)

Commented by Rasheed.Sindhi last updated on 21/Oct/21

T hank you dear \overset{?}{sir} / \overset{?}{madam} .

Certainly {there}^{'} s logical error and

I^{'} ll think over it .