epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

116
The Nuts and Bolts of Proofs, Third Edition
Then
^ppyfwf^.
This impUes that x^ = 3x. As x 7^ 0, the only solution is x = 3.
•
THEOREM 3.
There is no prime number larger than 12 million.
Alleged Proof. Let x be a number larger than 12 milUon.
Case 1. The number x is even. Then it is a multiple of 2, thus it is not
prime.
Case 2. The number x is odd. As x is odd, the two numbers a = (x + l)/2
and b = (x - l)/2 are both integers. In addition, 0 < ft < a. By performing
some algebra, one can prove that:
x = (a + b)(a - b).
As X is the product of two integers (namely, a + b and a — b), one can
conclude that x is not a prime number. Thus, there is no prime number
larger than 12 milhon. •
THEOREM 4. The graphs of the curves represented by the equations
y = x^ — X and y = \x — ^ have at most two points in common.
Alleged Proof 1. Graph both equations as shown here. The graph clearly
shows that there are two points in common. •
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