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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A paradox in differential calculus

Say I have a function $f=f(x,y)$ where $x,y$ are independent variables. Now, it is given that $p=x+y$. It can be shown that, since $x,y$ are independent, we get $$\frac{\partial p}{\partial x}=\frac{...
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0answers
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Convergence of sequence $a_0 := b$ and $a_{n+1} = 2^{a_n}$ in $\hat{\mathbb{Z}}$.

I have a question about the convergence properties of a sequence in $\hat{\mathbb{Z}}$, the completion of $\mathbb{Z}$. It is part of an exercise is due to this syllabus. I got confused somewhere. ...
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5answers
57 views

Little question about differentiation [duplicate]

My friend told me a wrong proof which involves differentiation, but I cannot point out where he has been wrong. $x^2 = x + x + x + … + x$ (total $x$ terms) ${\frac d {dx} x^2} = {\frac d {dx} (x + x ...
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1answer
40 views

Proof verification about inverses of linear mappings.

In order to prove the following statement: "Let $F: U\to V$ be a linear map, and assume that this map has an inverse mapping $G\colon V \to U$. Then $G$ is a linear map." In Serge Lang book he ...
18
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2answers
1k views

What's going on with this 5-line proof of Fermat's Last Theorem? [duplicate]

I'm reading a book on the Philosophy of Mathematics, and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author ...
31
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4answers
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Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.

In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\...
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1answer
111 views

Fake proof that $\frac{e^x-1}{e^x+1}=e^x$, via integrating $\operatorname{sech} x$ in two ways

We start with the integral: $$\int \text{sech}(x)dx$$ Method 1 \begin{align} \int \text{sech}(x)dx & = \int\frac{2}{e^x+e^{-x}}dx \\ &= \int\frac{2e^x}{e^{2x}+1}dx \end{align} Using the ...
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2answers
87 views

Integration by parts proof 1 = 0

Let's integrate $\int\frac{f^\prime(x)}{f(x)} dx$ by parts $$ \\ \mbox{ Let } dv= f^\prime(x)dx,u=\frac{1}{f(x)} \\ \mbox{ Then }v=f(x), du=-\frac{f^\prime(x)}{[f(x)]^2}dx \\ \mbox{ This implies }\int\...
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3answers
87 views

Proof Involving Imaginary Number: Where's the wrong one? [duplicate]

Here are the propositions: $$i=\sqrt{-1}$$ $$i^2=-1$$ $$(i)(i)=-1$$ $$\sqrt{-1}\sqrt{-1}=-1$$ $$\sqrt{(-1)(-1)}=-1$$ $$\sqrt{1}=-1$$ There's an error in the propositions above. I think it's in the ...
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1answer
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Failure of an elementary 'proof' of Fermat's Last Theorem?

Can someone explain to me why this does not constitute a proof of Fermat's Last Theorem, please? Basically, using something I've read online, it appears you can write an equation for $(a, b, c)$ to ...
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2answers
79 views

Help Debugging a Bogus Proof

We want to prove the standard fact that a smooth function $u :R^2 \to R$ with $ \nabla u = 0$ everywhere in some connected open set $ \Omega $ is constant in that set. I'm comfortable with the usual ...
3
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2answers
95 views

Is this logic of solving $\frac{0}{0}$ correct [duplicate]

I have seen such proofs many times and was unable to prove where it was wrong (not a math person-my bad). For example the following proof for $\frac{0}{0} = 2$ looks like it is proven correctly - but ...
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1answer
163 views

Fake proofs using matrices

Having gone through the 16-page-list of questions using the tag (fake-proofs), and going though Best Fake Proofs? (A M.SE April Fools Day collection) and https://en.wikipedia.org/wiki/...
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3answers
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What is the fallacy of this trigonometrical proof that $1=-1?$

I have this equation which I solved- $$\sin^4x-\cos^4x=1$$ $$\implies -\cos^4x=1-\sin^4x$$ $$\implies-\cos^4x=(1+\sin^2x)(1-\sin^2x)$$ $$\implies-\cos^4x=(1+\sin^2x)\cos^2x$$ $$\implies-\...
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5answers
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Unexpected result from Euler's formula

I am a bit confused with a result I get from Euler's formula: $e^{2\pi i} = 1$ $\sqrt[3] { e^{2\pi i} }= \sqrt[3]{ 1 }$ $(e^{2\pi i})^{\frac{1}{3}}= 1$ $e^{\frac{2}{3} \pi i} = 1$ This last ...
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1answer
55 views

Find the mistake of the following inductive proof: all algorithms have the same time complexity

I came across this problem: Find the mistake of the following inductive proof: Theorem: all algorithms have the same time complexity. Proof: (By induction on the number of algorithms.) The ...
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4answers
122 views

How do quaternions not show that $-1=1$? (Where is the proof wrong)

Given the rules of quaternions: $$ i^2=j^2=k^2=ijk=-1$$ could it not be used to show that $-1=1$? As follows: $$ijk=-1$$ $$ijk\cdot ijk=i^2\cdot j^2\cdot k^2=(-1)(-1)=1$$ $$i^2=-1$$ $$j^2=-1$$ $$k^...
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4answers
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Exponential decay 'proof' that $.\overline{9}\neq 1$?

I have doubts about $.\overline{9}$ being equal to 1 due to the following proof: To get a decimal containing $c$ 9's after the decimal point, the equation f(c) = $1-10^{-c}$ can be used. For ...
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3answers
109 views

What is the fallacy of this proof that $a=b$?

Let,you have an equation=$a^2-2ab+b^2$ This can be written in two ways- $$a^2-2ab+b^2\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space b^2-...
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3answers
3k views

Where does the gap come from? [duplicate]

Can anyone tell me please where does the gap come from? Thanks and sorry if the question is not exactly relevant, I just didn't know where else to ask.
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1answer
67 views

Direct proof that if mn is odd then m is odd and n is odd

I found the converse here, although that's not what I want. I have thought of a proof by contradiction and by contraposition, although I can't seem to figure out a way to finish a direct proof. $mn =...
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1answer
42 views

Whats wrong with this proof? (infinite sequences) [duplicate]

So I just watched a video where they explained that the sum of all natural numbers is $-1/12$. However, there was an interesting comment: ...
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5answers
95 views

Proof that $\{n\}$ is a Cauchy Sequence. Where is the fallacy?

We need to show that for every $\varepsilon>0$, $\exists N \in \mathbb R$ such that $n,m > N \implies |n-m|<\varepsilon$. $|n-m|<n+m$. So, if we make $n+m< \varepsilon$, the result ...
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3answers
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Why does the same equation have different results?

I bring the equation in (1) in order to ilustrate what I mean. Since (1.1) $12 - 6$ (1.2) $(4*3) - (2*3)$ (1.3) $(4-2) * (3)$ (1.4) $(2) * (3)$ (1.5) $6$ So... (2.1) $9.999... - 0.999...$ (2....
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0answers
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Question on an April Fool on Fourier transform

I have a question on this answer : Let $f(x) = 1$. It's easy to see that its Fourier transform is $0$ almost everywhere, so $\hat {\hat f}(x) = 0$. By the inversion theorem, $1 = 0$. I think ...
21
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6answers
4k views

Understanding Euclid's proof that the number of primes is infinite. [duplicate]

In Euclid's proof, if $p_1, p_2, \dots, p_n$ are the only primes then $p_1 \times p_2 \times \dots \times p_n + 1$ is not divisible by any of $p_1, p_2, \dots, p_n$ (because of some algebraic facts), ...
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1answer
73 views

Prove that if $A$ and $B$ are square matrices and $AB$ is invertible, then both $A$ and $B$ are invertible

I already know how to prove this using the definition of inverse and the associative property of matrix multiplication, but I was wondering if this would also be a valid proof. As $A$ and$ B$ are $n \...
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1answer
953 views

April Fools' Day Hoax: Fermat's Last Theorem

I read this answer to this question on MathOverflow, and I enjoyed reading the proof given in the linked paper, but... where is the mistake? I know nothing of the Mason-Stothers Theorem except its ...
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3answers
28 views

Find the Fallacy in the Factor Group Logic

The factor group for $(\Bbb{Z} \times \Bbb{Z})/\langle(2, 0)\rangle$ is clearly $\Bbb{Z} \times \Bbb{Z}_2$ because all of the elements are in the form of $a(1, 0)+b(0, 1)+\langle(2, 0)\rangle$ for $a \...
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2answers
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What exactly is the 'induction trap'

I've looked everywhere, and I've looked at a lot of examples. I don't quite understand what about the induction trap is so wrong. The most common example is the graph theory tree example (page 5 here: ...
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0answers
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Semigroup of a common face of two cones in a fan

Let $\Sigma$ be a fan, $\sigma_1, \sigma_2 \in \Sigma$, and $\tau = \sigma_1 \cap \sigma_2$ be a common face of the two cones. For a cone $\sigma$, denote the semigroup by $S_\sigma = \sigma ^\vee \...
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1answer
80 views

What is the proof for $\sqrt{-a}\times\sqrt{-b}\neq\sqrt{ab},\text{ where }a,b\in \mathbb{R}$

Having just learned about $i$ in my 10$^{th}$ grade classroom I'm interested in the proofs underlying the rules for algebraic manipulations with imaginary numbers; such an understanding will create a ...
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1answer
56 views

Fundamental theorem of Algebra in Complex Plane

Edit: I know that Euler had a method, which was incomplete, then Gauss proved the FTA. If somebody could show me what Euler did, it would be great. I think that's what I need. The Aim is to prove the ...
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0answers
34 views

Why do the integral and the partial sum agree for small $a$ and $m$?

Consider the following naive manipulations: \begin{align} \int_0^\infty \frac{e^{-x}}{1+ax}\:dx & = \int_0^\infty e^{-x}\frac{1}{1-(-ax)}\:dx\\ &= \int_0^\infty e^{-x} \left( \sum_{n=0}^\...
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4answers
167 views

Just got confused with what my friend asked (paradox and fake proofs). [duplicate]

Take $x^2=x+x+x+\cdots$ ($x$ times). Now differentiating both sides wrt $x$, we get: $$2x=x.$$ This means $x=0$ or $2=1$. How? Where did I go wrong?
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1answer
146 views

The three-coin-flip riddle

Is the following true (It seems obvious to me that it's not... but... a PhD in physics, Derek Abbott, seems to think others explanation at end of post): Someone flips 3 coins on the table, they are ...
1
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1answer
28 views

Birkhoff Ergodic theorem for two measures

Suppose $(X,\mathcal{B}, \mu, T)$ and $(X,\mathcal{B}, \nu, T)$ are both ergodic ppt. I'm a bit confused how the B Ergodic Theroem works since the LHS of the equation doesn't involve $\mu$ or $\nu$, ...
0
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1answer
102 views

Proving Infinite Nested Radical

This question asks to prove the limit of the infinitely nested radical. Now, I only have vague idea of what rigor means in proving something, but seeing my "answer" being radically different from ...
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1answer
58 views

How do I demonstrate that a logical error has been made?

I am grading an introduction to proofs course at my university. We have a question that reads something along the lines of: Prove that $$\forall \ n \in \Bbb N, [\exists \ a \in \Bbb N : n + 1 = 3a \...
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5answers
203 views

Infinite summation: $x+x+x+x+… =2$?

One of my favourite little math problems is this $x^{x^{x^{x^{...}}}}=2$ The solution to it is quite simple. An infinite tower of x's is equal to 2, and above the first x there is still an infinite ...
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3answers
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Can't find the flaw in the reasoning for this proof by induction?

I was looking over this problem and I'm not sure what's wrong with this proof by induction. Here is the question: Find the flaw in this induction proof. Claim $3n=0$ for all $n\ge 0$. ...
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1answer
42 views

What is wrong with this reasoning when calculating circle perimeter? [duplicate]

Looking at the following image, which was posted on the internet: Could someone tell me what is wrong? It seems true for the first 4 small images. But, when it comes to infinitesimal length, ...
0
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1answer
53 views

Contradiction in value of $\psi (1)$

What is the value of $\psi (1)$ ? If we take the definition in terms of derivative of Gamma function, we get $\psi (1) = \dfrac{\Gamma'(1)}{\Gamma(1)} = -\gamma$. But, if we consider the series ...
2
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1answer
67 views

What is wrong with the given proof? [duplicate]

Here is the proof they gave: Start with the statement $a = b$. Multiply both sides by $b$ to get $ab = b²$. Subtract $a²$ from both sides to get $ab − a² = b² − a²$. Factor the left and right sides of ...
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2answers
82 views

Alternative triangle inequality proof

I have looked everywhere for confirmation of this proof of the triangle inequality with no success. Prove the triangle inequality: $$\vert x + y \vert \leq \vert x \vert + \vert y \vert.$$ Proof: ...
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2answers
141 views

Another $1=2$ proof [duplicate]

So a friend shows me this : $x^4= x^2+x^2+ \cdots +x^2 $ ( i.e. $x^2$ added $x^2$ times) Now take the derivative of both side; $4x^3 = 2x + 2x + \cdots + 2x $; So $4x^3 = 2x^3 \cdots $(1) And ...
2
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4answers
59 views

Proof verification : every algebraic set is the union of finitely many irreducible algebraic subsets

I have found various proofs of the result but I have come up with something very different and I wonder whether it is a valid argument: Let $W$ be an algebraic set. Let $I=\mathcal{I}(W)$. We have $I=...
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2answers
34 views

Vacuous statements and explosion

So my understanding of vacuous statements is as follows: For any statement $P$, the statement $(\forall x \in \emptyset)(P(x))$. This can be argued as follows: Assume for contradiction $\neg [(\forall ...
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7answers
3k views

What is the flaw of this proof (largest integer)?

Let $n$ be the largest positive integer. Since $n ≥ 1$, multiplying both sides by $n$ implies that $n^2 ≥ n$. But since $n$ is the biggest positive integer, it is also true that $n^2 ≤ n$. It follows ...
0
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1answer
32 views

Covariance of dice tosses that result in 1 or 2 (fake proof)

Question: Consider n independent tosses of a $k$-sided fair dice. Let $X_i$ be the number of tosses that result in $i$. What is the covariance $\mathrm{cov}(X_1,X_2)$ of $X_1$ and $X_2$. \begin{...