Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

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Top new questions this week:

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...

(number-theory) (polynomials) (galois-theory) (irreducible-polynomials) (rational-numbers)  
asked by Dan Ismailescu 53 votes
answered by David 32 votes

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...

(linear-algebra) (matrices) (big-list) (applications)  
asked by Mike Miller 34 votes
answered by Huy 34 votes

What is the "fastest" increasing function that's useful in some area of math?

Context: I just completed the first quarter of an Intro to Real Analysis class, and while I was thinking about how some functions (like $x^2$) aren't uniformly continuous because they, roughly ...

(functions)  
asked by Michael Blakeman 27 votes
answered by MJD 33 votes

How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex ...

(number-theory) (cyclotomic-polynomial)  
asked by Bruno Joyal 23 votes

Is it OK to evaluate improper integrals this way?

Today in class we learned that when you have an improper integral like this one: $$\int_{-\infty}^\infty {f(x)} \: dx$$ you must split it before you do the limits (like so): $$\lim_{a \to \infty} ...

(calculus) (improper-integrals)  
asked by Roland Munsil 23 votes
answered by Bill Cook 29 votes

Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...

(real-analysis) (metric-spaces) (normed-spaces) (inner-product-space)  
asked by TaxxiDriver 21 votes
answered by rschwieb 24 votes

Unusual pattern in the distribution of odd primes

I have recently noticed an unusual pattern in the distribution of odd primes. Each one of the following sets contains approximately half of all odd primes: $A_n=\{4k+1: 0\leq k\leq ...

(number-theory) (prime-numbers)  
asked by barak manos 20 votes
answered by quid 24 votes

Greatest hits from previous weeks:

Interview riddle

On the Mathematics chat we were recently talking about the following problem @Chris'ssis had to solve during an interview : $$3\times 4=8$$ $$4\times 5=50$$ $$5\times 6=30$$ $$6\times 7=49$$ ...

(problem-solving) (puzzle)  
asked by Hippalectryon 41 votes
answered by Denis 98 votes

Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...

(soft-question) (big-list)  
asked by Calvin Lin 165 votes
answered by Luis Mendo 152 votes

Can you answer these?

Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$

(fractions) (continued-fractions)  
asked by Ehegh 12 votes

How to listen math lectures?

Many times, the lectures goes beyond the head and not easy to follow, (at least with me),mainly in the workshop/conferences, though the audience are eager to learn something from it. Also, most of ...

(soft-question) (education) (learning)  
asked by Inquisitive 7 votes

Examples of useful, insightful and interesting hand-waving

I am really amused by the answers to this question on "Most harmful heuristic" posed on MathOverflow, from which I've benefited a lot. However, it seems to me that some hand-waving may be really ...

(proof-strategy) (soft-question) (physics) (education) (big-list)  
asked by Dal 6 votes

New blog post:

Dealing with Risk

by nomen on Dec 15

Dealing With Risk Consider a small company which uses a million dollar machine as an essential part of its operations. Suppose that there is a 10% chance that the machine will break down and need ...

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