# Tagged Questions

For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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### Square root confusion: Why am I getting an answer if it doesn't work?

Alright, so I have $\sqrt{x-15} = 3-\sqrt{x}$. I first square both sides to get $x-15 = (3-\sqrt{x})(3-\sqrt{x})$ which simplifies to $x-15 = 9 -6\sqrt{x} + x$. I solved for $x$ and got $x = 16$, ...
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### Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
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### Is a matrix $A$ with an eigenvalue of $0$ invertible?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of ...
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### Is this proof of the infinitude of primes valid?

The current issue (May 2015) of the American Mathematical Monthly has a one-line proof that there are an infinite number of primes, and I don't see why it is correct. Here is the proof: If the set ...
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### The series $\sum_{n=1}^\infty\frac1n$ diverges

We all know that the following harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$$ diverges and grows very slowly!! I have seen many proofs of the result but ...
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### Is a brute force method considered a proof?

Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$". If we let a computer check every single member of $X$ and conclude that the ...
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### Question from Putnam '89: Primes of the form $101\ldots01$

I'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general input ...
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### Prove that zero multiplied by zero is equal to zero.

This is my proof: So, $0\cdot0=0$ And we know that $a-a=0$ By substitution, We have $(a-a)(a-a)=0$ Then by simplifying, $a^2-a^2+a^2-a^2=0$ and the we have $0-0=0$, Therefore, $0=0$. I am ...
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### What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?

This identity is solvable by the help of trigonometry identities, but I guess there is an interesting and simple geometry interpretation behind this identity and I can't find it. I found it when ...
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### Flaw or not flaw in Excel's RNG?

I have a question about my understanding of an article of B.D. McCullough (2008) about Excel's implementation of the Wichmann-Hill random number generator (1982). First, a bit of context The ...
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### If $N = q^k n^2$ is an odd perfect number and $n < q^{k+1}$, does it follow that $k > 1$?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $49$ known examples of even perfect numbers -- on ...
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### There's no cardinal $\kappa$ such that $2^\kappa = \aleph_0$

I am trying to prove that there is no cardinal $\kappa$ such that $2^\kappa = \aleph_0$ . My attempt: We suppose it exists. Since $\kappa<2^\kappa$, in particular, $\kappa<\aleph_0$. But ...
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### Proof: 1007 can not be written as the sum of two primes.

The claim is: 1007 can be written as the sum of two primes. We want to prove or disprove it. Edit: My professor provided this definition in his previous assignment: An integer $n \geq 2$ is ...
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### Alternative way to do this indefinite integral?

Problem : Solve $\displaystyle \int \frac{e^{3x}+1}{e^x+1}dx$. Attempt : Since we have \begin{align} \int\frac{e^{3x}+1}{e^{x}+1}dx & =\int\frac{e^{3x}}{e^{x}+1}dx+\int\frac{1}{e^{x}+1}dx \...
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### How many solutions for this equation?

$$\frac{x-4}{(x-1)} = \frac{1-4}{(x-1)}$$ Can someone tell me how many solutions are there for the above equation? MY APPROACH: I cross multiplied the equations and re-arranged to get a quadratic ...
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### Proving that there are infinitely many primes with remainder of 2 when divided by 3

I need to prove that there are infinitely many primes with remainder of 2 when divided by 3. I started out similarly to Euclid's classic proof of an infinite number of prime numbers: Suppose there is ...
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### No simple group of order $300$

So I've been trying to prove that there's no simple group of order $300$. This is what I did and I was wondering if it was enough. $|G|=2^2 \cdot 3 \cdot 5^2$. Suppose $G$ is simple. Then there ...
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### How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots$ $$n < \left(1+\frac{1}{\sqrt{n}} \right)^n$$ I tried with math induction but I failed.
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### Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
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### Function defined as a limit

Q. If $f(x)=\lim\limits_{n\to\infty}\dfrac{\log(2+x)-x^{2n}\sin(x)}{1+x^{2n}}$, then explain why the function does not vanish anywhere in the interval $[0,\pi/2]$, although $f(0)$ and $f(\pi/2)$ ...
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Problem. Let $f:A\to\mathbb{R}$ be a continuous function on $A$ and $g:B\to\mathbb{R}$ be a continuous function on $B$ such that $A\cap B=\emptyset$. Let $h:A\cup B\to\mathbb{R}$ be defined by, $$h(... 5answers 226 views ### I want to show that \int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3 I want to show that$$\int_{-\infty}^{\infty}{\left(x^2-x+\pi\over x^4-x^2+1\right)^2}dx=\pi+\pi^2+\pi^3 Expand $(x^4-x+\pi)^2=x^4-2x^3+2x^2-2x\pi+\pi{x^2}+\pi^2$ Let see (substitution of $y=x^2$)...
The ABC conjecture states that there are a finite number of integer triples (a,b,c) such that $\frac {\log \left( c \right)}{\log \left( \text{rad} \left( abc \right) \right)}>1+\varepsilon$, ...