150 Friends Problem - 3

Number Theory Level 5

x 4 = 4 x 3 y + 18 x 2 y 2 + 28 x y 3 + 15 y 4 \Large x^{4} = 4x^{3}y + 18x^{2}y^{2} + 28 x y^{3} +15 y^{4}

Find the number of positive integral solutions to the above equation.


The answer is 0.

Surya Prakash by Surya Prakash , 22, India

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2 solutions

Akul Agrawal
Oct 13, 2015

Factorizing the polynomial we get

( x + y ) ( 15 y 3 + 13 x y 2 + 5 x 2 y − x 3 ) = 0 (x+y)(15{ y }^{ 3 }+13x{ y }^{ 2 }+5{ x }^{ 2 }y-{ x }^{ 3 })=0

⇒ x + y = 0 O R x + y = 0.1404239... ( i r r a t i o n a l ) \Rightarrow \quad x+y=0\quad \quad OR\quad \quad x+y=0.1404239...(irrational)

RHS cannot be true as x and y are integers.

LHS cannot be true as x and y are positive.

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Shreyas Salvi
Sep 3, 2015

Solution for this Equation are in complex form so there r no positive integers for this equation

1 Helpful 0 Interesting 0 Brilliant 0 Confused

(1,-1) fit in the equations. Hence your solution is totally wrong...

Mayank Singh - 5 years, 7 months ago

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