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Questions tagged [fake-proofs]

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

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Why still people are searching for elementary proof of Fermat's Last Theorem?

I was searching in SE and Google for elementary proofs of Fermat's Last Theorem (FLT) and I found a lot of false claims about an elementary proof is found for FLT. I'm wondering: why still there are ...
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A strange proof of $\sqrt{-1}=1$ [duplicate]

I know that $\sqrt{-1}=i$,but I find a strange prove that it equals to 1. $$\sqrt{-1}=(-1)^\frac{1}{2}=(-1)^\frac{2}{4}=\sqrt[4]{(-1)^2}=\sqrt[4]{1}=1$$ Yes, it looks so strange and unreasonable. But ...
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6answers
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Proof that $0=1$? [duplicate]

I recently saw the following "proof" online, and couldn't pinpoint where the mistake was made: From a well known property, $$1+2+3+\cdots = -\frac{1}{12}.$$ Multiplying both sides by $-1,$ we get $$-...
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0answers
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Confusion about infinite product of fields

Consider $R=\Pi_{i\in \mathbb{N}} k$, where $k$ is a field. We know that its Spectrum $Spec(R)$ is quasi-compact. Moreover, by many answers already on this site, we know that the Krull dimension of $R$...
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3answers
78 views

Proof that the square root of a negative number is real.

So I stumbled upon this weird result when experimenting with fractional exponents. Suppose you have some negative, real number, for example -8. We know $\sqrt{-8}$ is not a real number.But $$\sqrt{-8} ...
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1answer
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What's wrong with my deformation retract?

I drew a deformation retract of the twice punctured genus 2-torus. It would seem that this should deformation retract to a bouquet of 3 circles. One the other hand, computing the Euler-...
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6answers
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Where is the logical flaw in solving this equation?

I ran across this equation... $\sqrt {2x+6}+4=x+3$ Without thinking, I solved for x in the following way: $\sqrt {2x+6}+4=x+3$ Subtract 4 from both sides. $\sqrt {2x+6}=x-1$ Square each side. ...
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2answers
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Fake proof that 1 = -1 (Can't understand the mistake) [duplicate]

$1 = \sqrt{1} = \sqrt{(-1)^2} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$. I know the mistake is here $\sqrt{(-1)^2} = \sqrt{-1}\sqrt{-1}$ because everything else seems right to me, but I don't understand ...
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Proof by deduction - implications

Currently trying to explain some maths to a friend. He has taken a statement $x^2 + 4 > 2x$ and tried to prove this is true for all $x$. His proof is $x^2+4>2x \Rightarrow x^2-2x + 4 > 0 \...
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1answer
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Locally free $\mathcal{O}_{X}$ modules are not projective

Let $(X , \mathcal{O}_{X})$ be a locally ringed space. I know that in general it is not true that locally free $\mathcal{O}_{X}$ modules are projective in the category of $\mathcal{O}_{X}$ modules (...
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3answers
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What's the error in my proof of the statement: The product of two irrational numbers is irrational?

Statement: The product of any two irrational numbers is irrational. Formally it can be written as: $$\Big(\forall x \forall y\Big)\,\Big(\big(x \notin \mathbb{Q} \wedge y \notin \mathbb{Q}\big) \to \...
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1answer
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Why do we need continuity of partial derivatives to prove differentiability?

I was reading the following theorem on Apostol's Mathematical Analysis (page 357): Assume that one of the partial derivatives $D_1f\dots D_nf$ exists at $\mathbf c$ and that the remaining $n−1$ ...
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0answers
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Finding an inconsistency in a seemingly flawless elementary proof of FLT

In an elementary "proof" (most likely, a wrong one) of Fermat's Last Theorem, I spotted an inconsistency I want you to confirm. I know, I am probably wasting my time doing this, but still, spotting a ...
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3answers
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Why the opposite direction of proving $\mathcal P(A) \cup \mathcal P(B) \subseteq \mathcal P(A\cup B)$ is wrong

The following proof is given to show that $\mathcal P(A) \cup \mathcal P(B) \subseteq \mathcal P(A\cup B)$ ($\mathcal P$ is the power set): Let $X \in (\mathcal{P}(A) \cup \mathcal{P}(B))$, by ...
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7answers
369 views

Elementary question about a fake-proof and greatest common divisors

I have a question to an excercise - for which I have a wrong solution - and I wanted to ask you to help me understand my thinking error. The excercise was as follows: Let $a, b, n \in \mathbb{N}$. ...
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3answers
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Where is the mistake in this “proof” of the inconsistency of ZFC?

This is a "proof" that ZFC is inconsistent, but I haven't found the mistake yet. Let $\{\varphi_n \colon n <\omega\}$ be an enumeration of all formulas in $L_{\in}$ with exactly one free ...
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5answers
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Starting with a false statement, how can one prove anything is true? [duplicate]

So I've been learning a bit of logic for class and heard that if you begin with a false statement, you can then prove anything to be true, however I don't entirely understand what this means or how to ...
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2answers
68 views

What went wrong in this fake proof?

So, I came up with this quite ridiculous fake proof that no nonnegative integers $(x,y)$ satisfy $x^2+y^2=5$. Clearly, $(x,y)=(1,2)$ satisfies the equation, so the proof is wrong; but somehow, I can't ...
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1answer
21 views

check if inequality is real

check if inequality is real: 30n < 2^n + 105 my solution proposal: n=1 30<107 true n=k 30k < 2^k +105 30(k+1) < 2^{k+1} + 105 Proof 30(k+1) < 2^{k+1} + 105 30k+30< 2^k +105+...
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What is wrong with this fake proof that subgroup of a cyclic group is cyclic?

Let $G$ be a cyclic group generated by $a$ and $H$ its subgroup. This is a proof by contradiction. Assume there is no $r$ in $H$ such that $\langle r \rangle = H$. If some $s$ in $G$ is not in $H$ ...
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1answer
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Why isn't Legendre's Conjecture resolved by the work done by Nair and Hanson in relation to Least Common Multiples?

I recently discovered the work done by Hanson and Nair. As I worked through Hanson's proof, an argument occurred to me regarding Legendre's Conjecture. Clearly, my argument is wrong. It is too ...
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1answer
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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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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
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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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0answers
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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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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
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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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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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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
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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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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
42 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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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
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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
64 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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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
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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
80 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
110 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
146 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
523 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 ...