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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
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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
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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
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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
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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

34 Answers 34

Let me recycle this.

$\phantom{*******}$

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

I have known the following for 45 years: in the Euclidian plane, every triangle is isosceles.

The false proof needs a handmade picture; take your pen, it's easy. Start from a triangle $ABC$. Draw the perpendicular bisector of $BC$, and the angle bisector from $A$. Let $I$ be their intersection (if it is not unique, you are done). Let $J$ be the projection of $I$ over $AB$, $K$ that over $AC$. Considering the right triangles $AIJ$ and $AIK$, we see that (lengths) $AJ=AK$, and that $IJ=IK$. Then looking at righttriangles $BIJ$ and $CIK$, we obtain that $BJ=CK$. We conclude that $$AB=AJ+JB=AK+KC=AC.$$

The falsity is that one of $J$ or $K$ is in the triangle, and the other one is out. Therefore one of the sums above (and only one) should be a difference.

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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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e is irrational.

Assume to the contrary that e were rational. Then e would be e-rational, a contradiction.

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It seems like mathematicians are completely and utterly incapable of understanding even the lowest level of humor (i.e. puns). –  Joseph Van Name Jun 16 '13 at 9:45
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I think it is more likely that at least 12 voters don't see puns as examples of interesting false proofs. In the event that you feel the absolute need to publicly communicate your displeasure, could you please find a way that doesn't involve insults? –  S. Carnahan Jun 18 '13 at 2:35
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Anyways. I am proud of this answer regardless of how people vote! –  Joseph Van Name Dec 18 '13 at 4:19

I'm fond of the following false proof of the Strong Law of Large Numbers. Let $X$ be a random variable with expected value $\mu$ and variance $\sigma^2$, and let $X_1, X_2, \dots$ be i.i.d. copies of $X$. Then $$\operatorname{Var} ( \frac{1}{n} \sum_{i=1}^n X_i ) = \frac{1}{n^2} \cdot n \sigma^2 = \frac{\sigma^2}{n} \rightarrow 0 \textrm{ as } n\rightarrow\infty $$ and since a random variable with variance 0 takes on a single value with probability 1, we must have $$\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{i=1}^n X_i = \mu \textrm{ almost surely.}$$ (It's a memorable heuristic reason to tell undergraduate probability students, even if not a true argument.)

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Theorem. $\int_0^\infty \sin x \phantom. dx/x = \pi/2$.

Poof. For $x>0$ write $1/x = \int_0^\infty e^{-xt} \phantom. dt$, and deduce that $\int_0^\infty \sin x \phantom. dx/x$ is $$ \int_0^\infty \sin x \int_0^\infty e^{-xt} \phantom. dt \phantom. dx = \int_0^\infty \left( \int_0^\infty e^{-tx} \sin x \phantom. dx \right) \phantom. dt = \int_0^\infty \frac{dt}{t^2+1}, $$ which is the arctangent integral for $\pi/2$, QED.

The theorem is correct, and usually obtained as an application of contour integration, or of Fourier inversion ($\sin x / x$ is a multiple of the Fourier transform of the characteristic function of an interval). The poof, which is the first one I saw (given in a footnote in an introductory textbook on quantum physics), is not correct, because the integral does not converge absolutely. One can rescue it by writing $\int_0^M \sin x \phantom. dx/x$ as a double integral in the same way, obtaining $$ \int_0^M \sin x \frac{dx}{x} = \int_0^\infty \frac{dt}{t^2+1} - \int_0^\infty e^{-Mt} (\cos M + t \cdot \sin M) \frac{dt}{t^2+1} $$ and showing that the second integral approaches $0$ as $M \rightarrow \infty$; but this detour makes for a much less appealing alternative to the usual proof by complex or Fourier analysis.

Still the double-integral trick can be used legitimately to evaluate $\int_0^\infty \sin^m x \phantom. dx/x^n$ for integers $m,n$ such that the integral converges absolutely (that is, with $2 \leq n \leq m$; NB unlike the contour or Fourier approach this technique applies also when $m \not\equiv n \bmod 2$). Write $(n-1)!/x^n = \int_0^\infty t^{n-1} e^{-xt} \phantom. dt$ to obtain $$ \int_0^\infty \sin^m x \frac{dx}{x^n} = \frac1{(n-1)!} \int_0^\infty t^{n-1} \left( \int_0^\infty e^{-tx} \sin^m x \phantom. dx \right) \phantom. dt, $$ in which the inner integral is a rational function of $t$, and then the integral with respect to $t$ is elementary. For example, when $m=n=2$ we find $$ \int_0^\infty \sin^2 x \frac{dx}{x^2} = \int_0^\infty t \frac2{t^3+4t} dt = 2 \int_0^\infty \frac{dt}{t^2+4} = \frac\pi2. $$ As a bonus, we recover a correct proof of our starting theorem by integration by parts:

$$ \frac\pi2 = \int_0^\infty \sin^2 x \frac{dx}{x^2} = \int_0^\infty \sin^2 x \phantom. d(-1/x) = \int_0^\infty \frac1x d(\sin^2 x) = \int_0^\infty 2 \sin x \cos x \frac{dx}{x}; $$ since $2 \sin x \cos x = \sin 2x$, the desired $\int_0^\infty \sin x \phantom. dx/x = \pi/2$ follows by a linear change of variable.

Exercise Use this technique to prove that $\int_0^\infty \sin^3 x \phantom. dx/x^2 = \frac34 \log 3$, and more generally $$ \int_0^\infty \sin^3 x \frac{dx}{x^\nu} = \frac{3-3^{\nu-1}}{4} \cos \frac{\nu\pi}{2} \Gamma(1-\nu) $$ when the integral converges. [Both are in Gradshteyn and Ryzhik, page 449, formula 3.827; the $\nu=2$ case is 3.827#3, credited to D. Bierens de Haan, Nouvelles tables d'intégrales définies, Amsterdam 1867; the general case is 3.827#1, from Gröbner and Hofreiter's Integraltafel II, Springer: Vienna and Innsbruck 1958.]

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+1: "Poof" is a great new term for an "incorrect proof," whether you intended it or not ;) –  David Corwin Mar 20 '13 at 18:37

Theorem: Every bounded differential function $f\colon \mathbb{R}\to \mathbb{R}$ is constant.

Proof. By assumption there exist real numbers $M,N$ such that
$$N\leq f(x) \leq M.$$ Taking derivatives we get $$0\leq f'(x)\leq 0.$$ Hence $f'(x)=0$ so $f$ is constant. QED

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Here's a nice false proof of the continuum hypothesis.

Consider the rational numbers $\mathbb{Q}$ as a totally ordered field. We can add an indeterminate $T_0$ and make it positive but infinitely small (i.e., smaller than positive any element of $\mathbb{Q}$), that is, order $\mathbb{Q}(T_0)$ by lexicographic order of the Laurent series expansion at $0$. Then we can add another indeterminate $T_1$ and make it positive but infinitely small (i.e., smaller than any positive element of $\mathbb{Q}(T_0)$). This process can be iterated transfinitely and we can add $\aleph_1$ indeterminates $T_\iota$ for $\iota<\omega_1$, each infinitely smaller than all the previous ones. The resulting field $K = \mathbb{Q}(T_\iota)$ has cardinality $\aleph_1$ as is easy to show. Now any positive sequence converging to $0$ in $K$ must be eventually constant because it has to cross uncountably many $T_\iota$. So any Cauchy sequence in $K$ is eventually constant. So any Cauchy sequence in $K$ is convergent. So $K$ is complete. But since $K$ contains $\mathbb{Q}$, it contains $\mathbb{R}$. So we have a set of cardinality $\aleph_1$ containing $\mathbb{R}$, which proves the continuum hypothesis.

(The error, of course, is simply that the notion of "completeness" is wrong and its use is nonsense. But if you tell it quickly enough, many people will fall for it.)

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I wouldn't say its "nonsense" -- there's a perfectly sensible notion of "completeness" and "completion" for ordered fields. It just isn't detectable via sequences in general; I'd say that's the real error here. –  Harry Altman Jan 15 '13 at 1:50

Here is an interesting false proof as to how to multiply $2 \cdot 2$. Taken from this link.

alt text


$\Large\textbf{Another example}$:

alt text

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An excelent example is the iscosceles triangle fallacy. Here is a link to it in wikipedia http://en.wikipedia.org/wiki/Mathematical_fallacy#Fallacy_of_the_isosceles_triangle

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In the definition of an equivalence relation $\sim$, the reflexivity of $\sim$ is redundant: Indeed, for any $x$, by the symmetric property we have $x \sim y$ implies $y \sim x$. By transitivity we have $x \sim y$ and $y \sim x$ imply $x \sim x$. Therefore, using only symmetry and transitivity, we obtain reflexivity.

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But this proves the result if there is at least one equivalence? –  David Corwin Mar 20 '13 at 18:02
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As Davidac says you only need that for any $x$ there exists at least one $y$ such that $x \sim y$. I set this as a homework question for my undergraduate groups course every year and the answers systematically ignore the necessary assumption –  Paul Levy Mar 20 '13 at 20:33

I can't remember where I first saw this: does anybody recognise it?

Let $I$ be the operator, from $C^0(\mathbb{R})$ to itself, which takes $f(x)$ to $\int_0^xf(z)dz$.

Since the exponential function $e(x)$ is its own derivative, we integrate both sides to get $e(x) = I(e(x)) + 1$. Regarding $1$ as the identity operator, we can rearrange to get $$(1-I)e(x) = 1,$$ and hence $$e(x) = \frac{1}{1-I}1 = (1 + I + I^2 + \cdots)1 = 1 + x + \frac{x^2}{2} + \cdots.$$

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True Theorem The symmetric groups (consisting of all permutations) on infinite sets of different cardinalities are not isomorphic.

False proof: The two groups have different cardinalities, since there are $2^\kappa$ many permutations of an infinite set of size $\kappa$, and $\kappa\lt\lambda$ implies $2^\kappa\lt 2^\lambda$. QED

See the question: Can the symmetric groups on sets of differing infinite cardinalities be isomorphic? for further information and a correct proof.

I find the false proof illuminating, since it shows the limitation of a naive treatment of the continuum function $\kappa\mapsto 2^\kappa$. It simply isn't necessarily the case that the two groups have different cardinalities, even though it is necessarily the case that they are not isomorphic.

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Theorem: Every totally disconnected set has the discrete topology.

Proof: Let $X$ be a totally disconnected set. If $X$ has only one element, the conclusion clearly follows. Otherwise, for distinct points $a, b \in X$, we have that {$a, b$} $\subset X$ is not connected. Therefore, {$a, b$} admits a separation; but the only way to write this as a disjoint union of nonempty sets is {$a$} $\cup$ {$b$}. Since this gives a separation, each of {$a$} and {$b$} is open. In particular, {$a$} is open for any $a \in X$; so $X$ has the discrete topology. Q.E.D.

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Well, the proof would prove more and be much simpler if, instead of looking at the subspace $\{a,b\}$, you just look at the subspace $\{a\}$. Now $\{a\}$ is obviously open, so every topological space whatsoever is discrete. –  Toby Bartels Jun 16 '12 at 15:36

$\pi$ is irrational : if $\pi=a/b$ is irreducible, and $a$ is divisible by an odd prime $p$, the series for $\sin \pi =\pi-\pi^3/6+\pi^5/120-\dots$ converges in the $p$-adics, and the limit is obviously not zero, absurd (if $a=2^n$, $n>1$ and the convergence is assured in the 2-adics, with the same contradiction).

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True story that I witnessed in a US precalculus class: the teacher told the class that $\pi$ was a rational number, since $\pi = C/d$, where $C$ is the circumference of a circle and $d$ is the diameter. Since $\pi$ can be written as a fraction, it is rational. This still makes me cringe to this day. –  John Engbers May 19 '12 at 18:28

I think that the history of this wrong proof of the Riemann hypothesis is pretty interesting:

http://www.math.columbia.edu/~woit/wordpress/?p=707

In the end, it motivated a paper by Bombieri and Lagarias

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.3791

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My favourites are "close" to formal false proofs in Coq.

1) In reply to a challenge by coq developer

Who can address this challenge: find a "simple" statement $T$ (simple in the sense that anyone with a minimal background in logics can understand) such that you can prove both $T$ and $\neg T$ in Coq.

Daniel Schepler solved it here. Daniel's proof was valid and passed coqchk, though it was not enough to prove False in Coq - Coq gave an "Universe inconsistency". AFAICT the proof encoded a paradox.

2) Damien Pous announced and gave link to code

There is a bug with vm_compute and values obtained from functors applications: using the attached code, I can produce an assumption-free proof of False, or Bus errors.

False proofs in Coq are difficult because Coq produces a "certificate" that can be checked for validity (if one doesn't check the certificate and is happy with the compiler as most people do, it is much easier).

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Timothy Chow's answer has a nice application. Let $n,x,y,z$ be natural numbers such that $x^n+y^n-z^n=0$. It follows that $e^{x^n+y^n-z^n}=1=e^i$ and the absurd $$1=(e^{x^n+y^n-z^n})^\pi=e^{i\pi}=-1.$$

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I always liked this proof, from the theory of Umbral Calculus developed by Rota (See "Combinatorics: The Rota Way", by Joseph Kung, Gian Carlo Rota and Catherin Yan, chapter 4.2).

Proposition: Let $(a_n)_{n\geq 0}$ and $(b_n)_{n \geq 0}$ be sequences. Then $$b_n=\sum_{k=0}^n\binom{n}{k} a_k \ \text{ for all } n \Longleftrightarrow a_n=\sum_{k=0}^n (-1)^{n-k}\binom{n}{k}b_k \ \text{ for all } n.$$

The heuristic proof use the notion of "raising and lowering subscripts and superscript". Raising subscripts at the left side we obtain $$b^n=\sum_{k=0}^n\binom{n}{k}a^k=(a+1)^n.$$ Hence, for all $n$, $$a^n=(b-1)^n=\sum_{k=0}^n (-1)^{n-k}\binom{n}{k}b^k.$$ Lowering exponents, we obtain the inverse relation.

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This doesn't look like a false proof. Rather, it's a proof that looks absurd at first glance but that can be made rigorous if you set up the right theoretical framework. Sort of like certain kinds of manipulations with divergent series, or arguments using infinitesimals, or the Dirac delta function. –  Timothy Chow Apr 23 '12 at 14:32

Theorem: All people have the same eye color.

Proof by induction: we prove the statement "All members of any set of people have the same eye color". This is clearly true for any singleton set.

Now, assume we have a set $S$ of people, and the inductive hypothesis is true for all smaller sets. Choose an ordering on the set, and let $S_1$ be the set formed by removing the first person, and $S_2$ be the set formed by removing the last person.

All members of $S_1$ have the same eye color, and also for $S_2$. However, $S_1 \cap S_2$ has members from both sets, so all members of $S$ have the same eyecolor. $\square$

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Some years ago, I came up with this false proof of the irrationality of $\pi$.

It suffices to prove that $x=\pi-3$ is irrational.

For real $y$ with $0\le y\lt1$, and positive integer $j$, define $d_j(y)$ to be the $j$th digit in the decimal expansion of $y$.

Let $r_1,r_2,\dots$ be an enumeration of the rationals in $[0,1)$. The $\it value$ of this enumeration is $n$ if $d_n(r_n)=d_n(x)$ and $d_j(r_j)\ne d_j(x)$ for $j\lt n$. If there is no such $n$, then the value of the enumeration is infinite. Note that if there is an enumeration of infinite value, then $x$ is irrational; it cannot equal any of the enumerated rationals, as it differs from the first rational in (at least) the first decimal place, from the second in the second, etc.

Note also that there are enumerations of arbitrarily large value. For, given any $n$, you can find $n$ rationals such that the first differs from $x$ in the first decimal, the second differs from $x$ in the second decimal, and so on, and then any enumeration that starts off with these $n$ rationals will have value greater than $n$.

Now, the set of all enumerations of the rationals can be partially ordered by value; if $E_1$ and $E_2$ are enumerations, then $E_1>E_2$ if the value of $E_1$ exceeds the value of $E_2$. By Zorn's Lemma, there is an enumeration maximal with respect to this order. This maximal enumeration cannot have a finite value --- as we have seen, there are enumerations of arbitrarily great finite value. So, it must have infinite value. So, $x$ is irrational.

An alternative use for this argument is to apply it to prove that $1/3$ is irrational, the contradiction with the known rationality of $1/3$ thereby establishing that Zorn's Lemma is false.

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Wouldn't it be easier (and pretty much equivalent) to prove that Zorn's Lemma is false by noting that it implies the existence of a largest natural number? –  Steven Landsburg Apr 23 '12 at 6:16
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Sure, but if you make it too easy you make it too obvious. Better to obscure the fallacy in lots of irrelevant verbiage. –  Gerry Myerson Apr 23 '12 at 12:51

A cavalry sergeant has 24 horses available which he needs to put on 6 carriages. So he needs to divide 24 by 6. He figures that 6 will go into 24 at least once, so he puts down a 1. Subtracting 6 from 24, he gets 18, and he remembers that 18/6=3. So he comes up with the answer 13.

After considerable difficulty with implementing his solution he consults his lieutenant. The lieutenant checks the calculation by evaluating 13*6:

3*6=18 1*6=6

Add them: 24.

Implementation of the result still remains elusive so they consult the colonel, who uses a different method to check. Write down 13 six times and add.

13

13

13

13

13

13

In adding this up, the colonel arrives at the following sequence of intermediate results: 3,6,9,12,15,18,19,20,21,22,23,24.

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I think something must have gotten lost in the typography--this doesn't make a lot of sense as it appears on my screen. –  Dylan Thurston Apr 22 '12 at 11:29
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This is incomprehensible as posted but begins to make sense after you've followed the link in Gerald Edgar's answer. –  Steven Landsburg Apr 22 '12 at 16:51
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Essentially the same proof is shown here: youtube.com/watch?v=Lo4NCXOX0p8 (an old Abbot & Costello sketch). –  Ketil Tveiten Apr 25 '12 at 11:11

Given any $x$, we have (by using the substitution $u=x^2/y$) $$ \int_0^1 {x^3\over y^2} e^{-x^2/y}\,dy = \biggl[x e^{-x^2/y}\biggr]_0^1 = x e^{-x^2}.$$ Therefore, for all $x$, $$\eqalign{e^{-x^2}(1-2x^2) &= {d\over dx}(xe^{-x^2})\cr &= {d\over dx} \int_0^1 {x^3\over y^2} e^{-x^2/y}\,dy\cr &= \int_0^1 {\partial \over \partial x} \biggl({x^3\over y^2} e^{-x^2/y}\biggr)\,dy\cr &= \int_0^1 e^{-x^2/y} \biggl({3x^2\over y^2} - {2x^4\over y^3}\biggr)\,dy.\cr} $$ Now set $x=0$; the left-hand side is $e^0(1-0) = 1$, but the right-hand side is $\int_0^1 0\,dy = 0$.

The main idea for this proof comes from an entry in Gelbaum and Olmstead's book Counterexamples in Analysis.

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$$e^i = (e^i)^{(2\pi/2\pi)} = (e^{2\pi i})^{1/2\pi} = 1^{1/2\pi} = 1.$$

I first saw this one many years ago, written on the wall of a bathroom stall in the Princeton University math department.

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Math departments have the best bathroom graffiti. –  Noam D. Elkies Jan 15 '13 at 4:51

Ma & Pa Kettle Math Lesson
YouTube

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I like this one, invented by T.Clausen in 1827: since $e^{2\pi i n}=1$ for all integers $n$, we have $e^{2\pi i n+1}=e$, which implies $e^{(2\pi i n+1)^2}=(e^{2\pi i n+1})^{2\pi i n+1}=e^{2\pi i n+1}=e$. Now expanding the square at the exponent gives $$e^{1-4\pi^2n^2+4\pi n i}=e$$ and after simplifying $$e^{-4\pi^2n^2}=1$$ for all $n$.

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How easily we forget that everything must be defined! –  some guy on the street Jun 16 '12 at 16:22

One usual "proof" of Leopoldt Conjecture is that $\mathbb{Z}_p$ is $\mathbb{Z}$-flat, hence the rank of the $p$-adic completion of the units of a number field has the same rank of the units themselves (which is Leopoldt Conjecture) because you can obtain the completion simply as $\mathcal{O}^\times\otimes\mathbb{Z}_p$.

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The Graham Pollak theorem is discussed at this link Combinatorial results without known combinatorial proofs . I came up with a nice short and incomplete proof of it. The tricky part for me was to realize it was incomplete. Follow the commentary if you want to see my "D'oh" moment. The induction started by taking an a,b complete bipartite subgraph of an (a+b) complete graph.

Gerhard "The Induction Looked So Pretty" Paseman, 2012.04.21

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