Top new questions this week:

What would base $1$ be?

Base $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be? Let's say we define Base $1$ to use: $\{0\}$. Because $10_2$ is equal to $010_2$, …

(number-systems)

asked by gangqinlaohu 34 votes answered by amWhy 27 votes

Why is every answer of $5^k - 2^k$ divisible by 3?

We have the formula $$5^k - 2^k$$ I have noticed that every answer you get from this formula is divisible by 3. At least, I think so. Why is this? Does it have to do with $5-2=3$?

(elementary-number-theory)

asked by iDivide 31 votes answered by Donkey_2009 78 votes

Is $\left( p+q\sqrt{3}\right) ^{1/3}+\left( p-q\sqrt{3}\right) ^{1/3}=n$ solvable in the natural numbers?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ …

(algebra-precalculus) (number-theory) (elementary-number-theory) (diophantine-equations)

asked by Américo Tavares 22 votes answered by Ian Mateus 5 votes

Predicting Real Numbers

Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to …

(recreational-mathematics) (puzzle) (axiom-of-choice)

asked by Jared 20 votes answered by mercio 6 votes

Does $\frac{x}{x}=1$ when $x=\infty$?

This may be a dumb question. I understand why $\frac{x}{x}$ when $x=0$ is undefined. This can causes errors if an equation is divided by $x$ without restrictions. $\frac{\infty}{\infty}$ is …

(calculus)

asked by monzie 19 votes answered by amWhy 28 votes

Why is $\frac{1}{\frac{1}{0}}$ undefined?

Is the fraction $$\frac{1}{\frac{1}{0}}$$ undefined? I know that division by zero is usually prohibited, but since dividing a number by a fraction yields the same result as multiplying the number …

(algebra-precalculus)

asked by Peter Olson 18 votes answered by Hagen von Eitzen 17 votes

About Euclid's Elements and modern video games

I just watched this video about Euclid's treatise the Elements. I got introduced to the postulates and a couple of propositions of book I. I really liked this video, I'm not sure if this is because of …

(geometry) (euclidean-geometry) (education) (math-software)

asked by Kasper 17 votes answered by Cam McLeman 4 votes

Greatest hits from previous weeks:

The Integral that Stumped Feynman?

In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods …

(real-analysis) (complex-analysis) (reference-request) (integral) (contour-integration)

asked by Argon 104 votes answered by Manoj Pandey 1 vote

How to check if a point is inside a rectangle?

There is a point (x,y), and a rectangle a(x1,y1),b(x2,y2),c(x3,y3),d(x4,y4), how can one check if the point inside the rectangle?

(analytic-geometry) (computational-geometry)

asked by Freewind 58 votes answered by lab bhattacharjee 75 votes

Can you answer these?

Model theory in terms of type spaces/Lindenbaum algebras

Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are …

(reference-request) (logic) (model-theory)

asked by KristianJS 5 votes

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to …

(numerical-methods) (powers) (continued-fractions) (tetration) (big-numbers)

asked by Vladimir Reshetnikov 12 votes

On the large cardinals foundations of categories

It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when talking about small categories, and perhaps at …

(set-theory) (category-theory) (large-cardinals)

asked by Asaf Karagila 7 votes