The Real Al - jabr by Shashvat Jayakrishnan

Number Theory Level pending

Solve the equation --- y 3 = x 3 + 8 x 2 − 6 x + 8 y^3 = x^3 + 8x^2 - 6x + 8 --- for integer values of 'x'and 'y'

x=0,y=2 ;; x=9,y=11 x=2,y=0 ;; x=11,y=9 x=0,y=9 ;; x=2,y=11 x=9,y=2 ;; x=0,y=11

Shashvat Jayakrishnan by Shashvat Jayakrishnan , 21, India

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1 solution

We have y 3 − ( x + 1 ) 3 = x 3 + 8 x 2 − 6 x + 8 − − − ( 1 ) { y }^{ 3 }-{ (x+1) }^{ 3 }\quad =\quad { x }^{ 3 }+8{ x }^{ 2 }-6x+8\quad ---(1)

Consider the quadratic equation 5 x 2 − 9 x + 7 = 0 − − − ( 2 ) . 5{ x }^{ 2 }-9x+7=0\quad ---(2).

T h e d i s c r i m i n a n t o f t h i e q u a t i o n i s Δ = 9 2 − 4 × 5 × 7 = − 59 < 0 A n d h e n c e e x p r e s s i o n n u m b e r ( 2 ) i s p o s i t i v e f o r a l l r e a l v a l u e s o f ′ x ′ W e c o n c l u d e t h a t ( x + 1 ) 3 < y 3 a n d h e n c e x + 1 < y . ( x + 3 ) 3 − y 3 = x 3 + 9 x 2 + 27 x + 27 − ( x 3 + 8 x 2 − 6 x + 8 ) = 2 x 2 − 18 x . W e c o n c l u d e t h a t x = 0 a n d y = 2 o r x = 9 a n d y = 11. The\quad discriminant\quad of\quad thi\quad equation\quad is\quad \\ \quad \quad \quad \quad \quad \Delta ={ 9 }^{ 2 }-4\times 5\times 7=-59<0\\ And\quad hence\quad expression\quad number\quad (2)\quad is\quad positive\quad for\quad all\quad real\quad values\quad of\quad 'x'\\ We\quad conclude\quad that\quad { (x+1) }^{ 3 }<{ y }^{ 3 }\quad and\quad hence\quad x+1<y.\\ \\ \quad { (x+3) }^{ 3 }-{ y }^{ 3 }={ x }^{ 3 }+9{ x }^{ 2 }+27x+27-({ x }^{ 3 }+8{ x }^{ 2 }-6x+8)=2{ x }^{ 2 }-18x.\\ We\quad conclude\quad that\quad x=0\quad and\quad y=2\quad or\quad x=9\quad and\quad y=11.

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This problem would have been better if you did not make it Multiple Choice. Trivially, ( x , y ) = ( 0 , 2 ) (x,y)=(0,2) is a solution, and only one of the choices reflect that.

Daniel Liu - 6 years, 10 months ago

FYI, To type equations in Latex, you just need to add the brackets \ (   \ )  around your math code, as opposed to all of the text. In this way, you don't have to use \quad all the time. I've edited the start of your solution as a reference.

Calvin Lin Staff - 6 years, 10 months ago

Can you check your first line? I believe that you have not done the subtraction yet.

Calvin Lin Staff - 6 years, 10 months ago

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