TITLE: What are some classic fallacious proofs?
QUESTION [20 upvotes]: If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:

 Let $a = b$.
 Then $a^2 = ab$
 $a^2 - b^2 = ab - b^2$
 Factor to $(a-b)(a+b) = b(a-b)$
 Then divide out $(a-b)$ to get $a+b = b$
 Since $a = b$, then $b+b = b$
 Therefore $2b = b$
 Reduce to $2 = 1$

As @jan-gorzny pointed out, in this case, line 5 is wrong since $a = b$ implies $a-b = 0$, and so you can't divide out $(a-b)$.

REPLY [6 votes]: The odd number $N = 198585576189 =
3^2 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 22021$ has an interesting property—it is perfect:
$$\sigma(N) = (1 + 3 + 3^2)(1 + 7 + 7^2)(1 + 11 + 11^2)(1 + 13 + 13^2)(1 + 22021) = 397171152378 = 2N$$
Now, where is the catch? (This one was found by René Descartes. It is also the only known odd number to have this property.)

 We pretend that the number $22021 = 19^2 \cdot 61$ is prime.