The discussion revolves around finding a relationship between the parameters \(a\) and \(b\) in the context of two equations involving a variable \(x\), where \(x\) is specified as a negative integer and \(y\) as a positive integer. The focus is on deriving a relative equation and identifying pairs of \((x,y)\).
The discussion includes multiple inquiries and responses, but there is no consensus on the relationship between \(a\) and \(b\) or the pairs of \((x,y)\). The co-primality question also introduces a separate line of inquiry that remains unresolved.
The discussion does not clarify the assumptions regarding the values of \(a\) and \(b\), nor does it resolve the mathematical steps needed to derive the relationship between them. The implications of the negative integer condition for \(x\) and the positive integer condition for \(y\) are also not fully explored.
Albert said:$y=x^3-ax^2-bx---(1)$
$y=ax+b---(2)$
$a,b\in R$
$x$ is a negative integer, $y$ positive integer
(1) find the relative equation between $a$ and $b$
(2)pair(s) of $(x,y)$
kaliprasad said:we have $y=x^3- ax^2-bx = x^3 - x(ax+b) = x^3-xy$ using (2)
or $y+xy = x^3$
or $y = \frac{x^3}{1+x}$
now $x^3$ and (1+x) are co-primes and 1 + x is not a factor of $x^3$ unless $1+x = \pm 1$
and $1+x=1$ gives x = 0 which is not admissible
so $1+x = -1$ hence $x = -2 $ and $y=8$ so this is the solution as it satisfied x -ve and y + ve
putting in (2) we get
8 = -2a + b or $2a-b = 8$ is the relationship between a and b
Fermat said:How do you know $x^3$ and $1+x$ are co-prime?
kaliprasad said:because $x^3+1 = (x+1)(x^2-x + 1)$
so $x^3 = (x+1)(x^2-x+1) + 1$
so $GCD(x^3, x + 1) = GCD(- 1,x +1) =1$ or -1 if you like it and hence co-primes unless $x +1 = \pm 1$