# Examples of interesting false proofs

According to Wikipedia False proof

For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way.

The Wikipedia page gives examples of proofs along the lines $2=1$ and the primary source appears the book Maxwell, E. A. (1959), Fallacies in mathematics.

What are some examples of interesting false proofs?

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Is this a duplicate? –  Bruce Westbury Apr 21 '12 at 17:19
the answers to this will turn out to replicate many of the responses to Gowers' famous question on "false beliefs", so I am not so sure if this question should remain open. –  Suvrit Apr 22 '12 at 5:46
A false proof is not the same as a false belief. One can read a false proof, know for certain that the conclusion is false (so there is no false belief), and still have trouble pinpointing the error. –  Steven Landsburg Apr 22 '12 at 15:36
Indeed, a false proof is not the same as a false belief, and at no point did I imply that! But I mentioned Gowers' question, because the top answer's "false proof" (Cayley-Hamilton) also occurred there as one of the answers. (Believing a "false proof" to be true, is a "false belief", and because of that, there is a strong chance of intersection between the two questions) :-) –  Suvrit Apr 22 '12 at 22:27
I'm surprised no one has mentioned Stallings's false proof of the Poincare Conjecture, in his paper "How Not to Prove the Poincare Conjecture". –  Steve D Apr 30 '12 at 22:13

In S. Bosch's Algebra, exercise 3.4.2 is to find an error in the following existence proof of an algebraic closure of a field $K$ (my translation):
"Consider all algebraic extensions of $K$. Since for a totally ordered (w.r.t. inclusion) family $(K_i)_{i \in I}$ of algebraic extensions of $K$, the union $\bigcup_{i \in I} K_i$ is an algebraic extension of $K$, Zorn's lemma shows the existence of a maximal algebraic extension, i.e. of an algebraic closure of $K$."

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I think nobody point to these interesting false proof:

Let $i=\sqrt{-1}$ be the complex number.

$1)$ $1=\sqrt{-1\times-1}=\sqrt{-1}\times\sqrt{-1}=i\times i=-1$.

$2)$ We know that $x^\frac{2}{6}=x^\frac{1}{3}\Rightarrow (\sqrt{x^2})^\frac{1}{6}=(\sqrt{x})^\frac{1}{3}$. Now, let $x=-1$ and so we have: $$(\sqrt{(-1)^2})^\frac{1}{6}=(\sqrt{-1})^\frac{1}{3}\Rightarrow1=-1.$$

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Doron Zeilberger proved that P is equal to NP

Abstract: Using 3000 hours of CPU time on a CRAY machine, we settle the notorious P vs. NP problem in the affirmative, by presenting a “polynomial” time algorithm for the NP-complete subset sum problem. Alas the complexity of our algorithm is $O(n^{10^{10000}})$ (with the implied constant being larger than the Skewes number).

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This is 11 days too early. –  Noam D. Elkies Mar 21 '13 at 15:24
@Noam: Actually, 11 days minus 4 years too early. –  Lee Mosher Mar 21 '13 at 15:29