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Use proof by contradiction to show that if a, b and c are odd integers then ax2 +bx+c = 0 has no solution in the set of rational numbers.
(You may use standard parity results of the form "odd x odd = odd", "even + odd = odd", etc.)
Don't understand how to prove this :(
Assume that there is a rational solution x. How can x be written? Substitute this form of x into the equation and convert it into an equation that uses addition and multiplication only. Consider several cases when the variables occurring in the equation are even or odd. Only one of those cases is possible. But then the representation of x chosen in the beginning was not optimal in some sense.
Suppose there is a rational root in the formwhere p, q are relatively prime and
. Then
Note that a,b,c are all odd. Can you find a parity contradiction here?
I am new to the forum and am an amateur, so I may have oversimplified.
Move c to the right side of the equation, factor the left side by x, thus x has to divide c, which is odd, therefore x must be odd. Since a and b are odd, x(xa + b) is even, and we have a contradiction.
What does it mean for a rational number to divide an integer?Quote:
Originally Posted by OldDude
a,b,c odd => a = b = c = 1 (mod 2).
thus, mod 2:
ap2 + bpq + cq2 = 0 becomes:
p2 + pq + q2 = 0 (mod 2)
and since mod 2, p2 = p (a specific instance of fermat's "little theorem" for p = 2, but it's easy to verify directly 02 = 0, and 12 = 1, and (mod 2), 0 and 1 are "all there is"),
p + pq + q = 0 (mod 2)
if p = 0, q = 1 we have:
0 + 0 + 1 = 0 (mod 2) can't happen.
if p = 1, q = 0, we have:
1 + 0 + 0 = 0 (mod 2) nope. never.
if p = q = 1, we have:
1 + 1 + 1 = 0
0 + 1 = 0 sorry, wrong number.
so the only legitimate possibility is p = q = 0 (mod 2):
0 + 0 + 0 = 0 (mod 2). we have a winner!
but wait! if p = q = 0 (mod 2), this means both p and q are EVEN.....so gcd(p,q) can't possibly be 1 (it has to be at LEAST 2).
(arithmetic "mod 2" IS the "arithmetic of parity").