Show that, regardless of what integers are substituted for x and y, the expression
x5 - x4y - 13x3 y2 + 13x2 y3 + 36xy4 - 36y5
is never equal to pq, where p and q are prime numbers.
The expression can be factored as (x-3y)(x-2y)(x-y)(x+2y)(x+3y).
If (x,y) are both even, every factor is even.
If (x,y) are both odd, 3 factors are even.
So (x,y) have opposite parity and 3 of the factors = 1 while the remaining factors are prime.
If both (x-3y) and (x+3y) = 1 then (x,y)=(1,0) and the expression evaluates to 1. The same result obtains when considering (x-2y) and (x+2y).
So at most one of each pair = 1 and therefore (x-y)=1. But this fact considered with whichever of (x-3y) and (x+3y) = 1 leads to y=0.
Since this exhausts the possibilities the proof is complete.
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Posted by xdog
on 2019-06-01 14:10:05 |
No Subject