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Questions tagged [fake-proofs]

Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This tag is the appropriate place to ask "Where is the proof wrong?" when encountering such proof.

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What's wrong with this induction? (Runtime analysis of standard polynomial long division)

In my studies for my Bachelor's thesis, I've gone through a runtime analysis of plain vanilla polynomial long division, i.e. I wanted to prove the statement: Let $f,\,g \in F[X] \land g\neq 0$ where $...
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2answers
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Why does my proof fail to show the logical equivalence of (∀x)(Fx v Gx) ⊢ (∀x)Fx v (∀x)Gx?

Apparently (∀x)(Fx v Gx) is not equivalent to (∀x)Fx v (∀x)Gx, however I seem to be able to prove it syntactically: (∀x)(Fx v Gx) ⊬ (∀x)Fx v (∀x)Gx (1) (∀x)(Fx V Gx)-----premise (2) Fa v Ga----------...
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0answers
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What's the fallacy in $i^{-1} = i$ [duplicate]

We need to express $i^{-1}$ as $a+bi$ where $a,b \in \mathbb{R}$. There are a lot of ways to simplify it. One can easily see that $i^{-1} = i^3= -i$, for example. However, this is not the approach ...
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1answer
45 views

Where does this derivation of the Fourier Series for csc(x) go wrong?

In this post, the following derivation for the Fourier series of csc(x) is given: \begin{align} \csc x &= \dfrac{1}{\sin x}\\ &= \dfrac{2i}{e^{ix}-e^{-ix}}\\ &= \dfrac{2ie^{-ix}}{1-e^{-...
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Feedback needed on Proof using Strong Induction Proof $S(x) = 4 S(\frac{x}{2}) + x$

Given: $S(x) = 4S({\lfloor}x/2{\rfloor}) + x, S(1)=1$. Claim: $S(x)$ equal to $O(x)$ as asymptotically. (Which is obviously not true.) Strong Induction Proof also given as: Base Case:(BC) when $x = ...
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28 views

Wrong intuition with partial derivatives.

So I noticed I have some wrong intuition about partial derivatives and I would appreciate someone correcting me why I am wrong. Say we have a function $f(x,y)=2\sqrt{xy}$. Clearly $f_x(x,y)$ ...
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1answer
7k views

What is the Todd's function in Atiyah's paper?

In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?
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3answers
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Why are cases required in proofs? i.e. in the inequality $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|}$?

So there is a proof of the inequality needed: $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|} $ , where $x$,$y$ $\geq$ $0$ After squaring both sides: $(|\sqrt{x}-\sqrt{y}|)^2 \leq (\sqrt{|x-y|})^2$ $x + y ...
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2answers
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Let $C$ be compact sets. $\bigcap_{i \in K} (C_i)=\emptyset \Rightarrow \exists F \subseteq K \; \text {finite} | \bigcap_{i \in F} (C_i) = \emptyset$

I made a mistake while proving the theorem in the title, however, I don't see where. Here's the theorem: Let $C$ be closed and limited subsets of $\Bbb{R}$. Then $\bigcap_{i \in K} (C_i) = \...
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3answers
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Proof that 2=1 using differentiation [duplicate]

We can express a square number as the repeated addition of that number in this manner: $1^2 = 1$ $2^2 = 2 + 2$ $3^2 = 3 + 3 + 3$ Generalising this, we get: $x^2 = x + x + x...$ $x$ $times$ If we ...
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1answer
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Understanding a mistake regarding removable and essential singularity.

It is a well known fact that $0$ is an essential singularity of the function $e^{1/z}$ therefore it is not essential nor a pole. On the other hand we know the Rieamann's continuation theorem which ...
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54 views

Why is this counterexample to an established result wrong?

It is a well known fact that, for a real function $f$, if $f'(x)$ has a limit as $x\to x_0$, then $f$ is differentiable at $x_0$. In fact, there has been some questions on this site about proving this ...
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0answers
95 views

Modern Mathematical Crankery: A List.

I'm sorry if this question is off topic. I'd also like to apologise if this is deemed too broad or otherwise a poor question. The Question: I'm interested in examples of modern mathematical ...
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1answer
31 views

Proof of the formula for area of a polar function.

I was wondering what i am doing wrong in my proof of the formula $$A=\int r^2/2$$ First of all we know, given parametric equations $x=f(t)$ and $y=h(t)$ the area represented by that parametric curve ...
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3answers
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Where is wrong with this fake proof that Gaussian integer is a field?

The Gaussian integer $\mathbb{Z}[i]$ is an Euclidean domain that is not a field, since there is no inverse of $2$. So, where is wrong with the following proof? Fake proof First, note that $\...
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1answer
41 views

Where is the error in this proof?

Let $\mathfrak{g}$ be a Lie Algebra. Then isn't necessarily true that all vector spaces $V \subset \mathfrak{g}$ are a Lie subalgebra (it is easy to construct an example that this fails). Now, weird ...
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0answers
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Is this proof of $0 = \infty$ just a mathematical joke? [duplicate]

Is this "proof" just a mathematical joke or might there be some deeper truth in it, eventhough the theorem is obviously false? Definition: Let a regular $n$-gon be a geometric figure $f^d_n$ that ...
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1answer
76 views

How can adding new numbers overflow $\mathbb{N}$ in Cantor's diagonal argument?

I've been thinking and asking around about this for a while. So I think Cantor's diagonal argument basically said that you can find one new number for every attempted bijection from $\mathbb{N}$ to $\...
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1answer
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Error in “proof” that every quotient of PID is PID

I know that if $A$ is a PID and $I$ an ideal, then $A/I$ need not be a PID, since it's not even a domain unless $I$ is prime. However, I can't quite seem to find my mistake in the following "proof" ...
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1answer
58 views

Is this proof of Cauchy about limits valid?

I've read in some places that the following proof from Cours d'Analyse of Cauchy is not correct, but I can't find the mistake. Theorem: if the difference $f(x+1)-f(x)$ converges towards a certain ...
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2answers
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Writing a Fake Proof for Real Analysis

mathematics community! I'm teaching a course in Real Analysis soon, and one thing I wanted to include were a few "fake proofs" for my students to evaluate. The research I've done hasn't turned up any ...
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Bounding 2D Brownian Motion expected barrier hitting time of a square

I have been doing a computation but I am making a mistake somewhere and cannot figure out where. The question I have is: where am I making my mistake? I will run through my reasoning: Consider ...
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3answers
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Where did I go wrong with my odd proof that $\frac{3dx}{3x} = \frac{5dx}{5x} \iff 3=5$?

I don't know where I went wrong, but it's interesting for me. Please check where my fault is! It is obvious that the below equation is correct: $$\frac{3dx}{3x}=\frac{5dx}{5x}$$ $$u=3x$$and$$v=5x$$ $...
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1answer
57 views

A direct way to an inequality : Ferrari's identities

I would like to submit a recent answer that I gave (and I have deleted) where someone tolds me that I was "total wrong" this is the following : Begin Prove that $$\frac{x^2+y^2+z^2}{2}\geq (\alpha\...
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3answers
75 views

Which one of these geometry proofs are incorrect?

One proof proves an angle to be 90º, and the other proof proves the very same angle to be 60º. I've looked over both proofs several times but I can't find what the error in either one of them is. And ...
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1answer
103 views

Can an error be found in this proof of Gödel's incompleteness theorem?

Can you find a division by zero error in the following short proof of Gödel's incompleteness theorem? First a little background. $\text{G($a$)}$ returns the Gödel number of the formula $a$. As in ...
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1answer
143 views

ln(2) contradiction

$\ln2\approx.693$, according to my calculator. It can be written as the infinite sum $$1-\frac12+\frac13-\frac14+\frac15-\frac16+\frac17-\frac18+\frac19-\frac1{10}\dots$$ Rearranging this infinite ...
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0answers
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False Proof Identification [duplicate]

I'm having some trouble identifying where specifically the following proof is incorrect: Statement: Given any positive integer $n$, every n people have the same name. "Proof:" We prove by induction. ...
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4answers
513 views

How does this proof of the regular dodecahedron's existence fail?

On Tim Gowers' webpage he has an example "proof" of the regular dodecahedron's existence which he claims contains a flaw. He writes Of course, I have not written the above proof in a totally ...
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9answers
187 views

Fake proof that $1$ is the solution of $x^2+x+1=0$

So I have this false proof and I am honestly confused why this is happening. Consider $x^2+x+1=0$, then $x+1=-x^2$. Now by simply dividing the equation by $x$ we get $x+1+1/x=0$. Substituting $x+1=-...
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2answers
751 views

Finding mistake in proof that all numbers are even [closed]

Claim: The numbers $0,1,2,3,\dots$ are all even. Proof: We use strong induction to prove the statement '$n$ is even' for $n=0,1,2,3,\dots$ Base case: $n=0$ is an even number, hence the ...
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4answers
696 views

Adding infinite and finite numbers: why doesn't 0=1?

Okay, so, $$\infty + 1 = \infty$$ subtract infinity from both sides. $$1=0$$ At first I thought, duh, $\infty \neq \infty+1$, but, now, I'm just more confused because my brother rephrased it in ...
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3answers
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Finding a mistake in this 'proof' that $1 = 2$. [duplicate]

I have some idea of where the mistake in this 'proof' may be, but can't quite formalize. We start with the trivially correct statement, $1 - 3 = 4 - 6$. Then, completing the square on the LHS, we add ...
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2answers
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About an exercise in Rudin's book

In the book of Walter Rudin Real And Complex Analysis page 31, exercise number 10 said: Suppose $\mu(X) < \infty$, $\{f_n\}$ a sequence of bounded complexes measurables functions on $X$ , and $f_n ...
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3answers
103 views

What is the correct proof that when $\epsilon$ decreases (in a limit) then the largest $\delta$ should also decrease?

In my search for this statement I wrote this question (a few years ago) When $\delta$ decreases should $\epsilon$ decrease? (In the definition of a limit when x approaches $a$ should $f(x)$ approach ...
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2answers
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Uniform limit of functions with intermediate value property has intermediate value property? Where is the error in this proof?

This is the question that I posed myself and set out to solve before I came to MSE: Suppose that $\{f_n\}$ is a sequence of real-valued functions defined on $[a,b]$ such that each function has the ...
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3answers
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Lesser-known fake proofs [closed]

I believe I have seen almost every elementary bogus proof of $0=1$ or $1=2.$ I am wondering if there are any lesser-known "proofs" where the error is harder to spot.
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$ \operatorname{dist}(w,A) \ge \operatorname{dist}(w, \partial A)?$ What is the mistake in this proof?

Suppose $(X,d)$ is a metric space, $w \in X$ amd $A \subseteq X$. $\partial A$ refers to the set of boundary points of the set $A$ defined as $\{ x \in X~|~ \operatorname{dist}(x,A)= 0 = \...
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1answer
38 views

Wrong proof for the variance of a sum of normally-distributed variables?

I'm reading the book "Introduction to Error Analysis" by John R. Taylor. The author is discussing the probability distribution of a sum of two normally-distributed random variables, and wants to show ...
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2answers
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On The “Commutative” Properties Of Subtraction And Division

Can someone explain to me why subtraction and division is both commutative? The reason I believe that they are commutative are as follows: $$ 3-2 = 3+(-2) $$ $$ 3+(-2) = (-2)+(3) $$ I've read a ...
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4answers
188 views

How to prove that $-x$ is not equal to $x$ just because they yield the same result when in $x^2$

I know how incredibly stupid this sounds, but bear with me. Let's take any random $x$, say $3$, and any random $-x$, say $-3$. Let's plug it into $x^2$. They will both give the same result! I know ...
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0answers
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Questions about the statement “Every number can be specified by less than twenty words.”

This is really an interesting question, though I do not know how to word it in a mathematical way. I am glad if one can help me to reword it mathematically. A friend of mine comes up with this ...
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1answer
32 views

Erroneous proof that derivative tending to $\infty$ implies that the function is not uniformly continuous.

I have a proof for the following statement (which I know is wrong): Let $f:(a,b)\rightarrow \mathbb{R}$ be a differntiable function such that $\underset{x\rightarrow a^+}{\lim} \vert f'(x) \vert=\...
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2answers
73 views

Proof that the set of Natural numbers are equal to the Real numbers [duplicate]

Obviously the proof is flawed in some way, but I have been struggling to find out where the flaw is. I am self taught and fairly new to set theory so the flaw may be obvious (sorry): The idea is that ...
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1answer
65 views

Topology of almost everywhere convergence - What's wrong with my construction?

I understand that the a.e. convergence is not topologizable, as there is contradiction, for instance with the fact that $L^p$ convergence does not imply a.e. convergence. So I am asking what is ...
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1answer
216 views

Is this considered a disproof of $1+2+3+4\ldots=-\frac{1}{12}$? [closed]

The series "$1+2+3+4\ldots=-\frac{1}{12}$" didn't seem to make sense to me as it breaks clear rules of series (but I am yet to research if the rules it broke aren't breakable(doesn't work if broken) ...
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3answers
96 views

Find all $x$ such that $4^{27}+4^{1000}+4^{x}$ is a perfect square.

Let,$4^{27}+4^{1000}+4^{x}=n^2$ $4^{2}(4^{25}+4^{998}+4^{x-2})=n^2$ LHS is multiple of 4.$n$ is also multiple of 4 Let $n=4a _1$. $4^{2}(4^{25}+4^{998}+4^{x-2})=16.a_1^2$ $4^{25}+4^{998}+4^{x-2}=...
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3answers
92 views

Bogus Proof by Strong Induction

So here is a bogus proof. Let $$P(n) ::= \forall k \leq n, a^k =1, $$ where k is a nonnegative valued variable. Base Case: $ P(0) $ is equivalent to $a^0 = 1$, which is true by definition of $...
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0answers
66 views

All positive integers $n$ are equal

Is my proof correct? I have to prove that: All positive integers $n$ are equal $P (n)$: All numbers in a set of $n$ positive integers are equal Base: $P (1)$ is true Inductive hypothesis: Suppose ...
3
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2answers
50 views

Misunderstanding Caley-Hamilton Theorem - Characteristic Polynomial in the Standard/Factorized Form

so my question is about the Caley-Hamilton theorem. Consider the following Matrix A. $$A =\begin{pmatrix} -1 & 0 & 4 \\ 2 & -1 & 0 \\ 3 & 2 & -1 \end{pmatrix}$$ The ...