Let me first clarify that I am not expecting the titular question to have a single well-defined answer (or it would probably have to be a useless one like “the topos freely generated by the followi...

Most of us know the Jacobian conjecture. Here's a version below for fixed positive integers $d$ and $n$: $J(d,n)$: If $f: C^n \rightarrow C^n$ is a polynomial map of degree $d$, and if the Jacob...

Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation it stays a convex polytope, it stays a combinatorial dodecahedron (i.e. ...

Here’s kind of a strange problem with an interesting solution: find a function _f_ such that the derivative of _f_ equals the inverse of _f_ for all positive _x_. _f_ ′(_x_) = _f_ −1(_x_) This is a differential equation, but a very unusual one, one that cannot be solved using any of the techniques taught in a class on differential equations. The unique solution is _f_(_x_) = φ(_x_ / φ)φ where φ is the golden ratio. What an unexpected appearance of the golden ratio! The problem was proposed by H. L. Nelson and solved by A. C. Hindmarsh. See The American Mathematical Monthly, Vol. 76, No. 6 p. 696.

The satellites of Uranus orbit in a low-eccentricity, equatorial plane that is tilted by 98° relative to the solar system—a geometry that mirrors U...

Background. A set $X$ is Dedekind-finite, or D-finite, if every injection $\iota:X\to X$ is surjective. In ${\sf ZF}$, if $X$ is a D-finite set, its powerset ${\cal P}(X)$ is not necessarily D-fin...

Here’s a curious theorem I stumbled across recently [1]. Take an integer _N_ which is not a multiple of 10. Then there is some multiple of _N_ which only contains the digits 1, 2, 3, 4, and 5. For example, my business phone number 8324228646 has a couple 8s and a couple 6s. But 6312 × 8324228646 = 52542531213552 which contains only digits 1 through 5. For a general base _b_ , let _p_ be the smallest prime factor of _b_. Then for every integer _N_ that is not a multiple of _b_ , there is some multiple of _N_ whose base _b_ representation contains only the digits 1, 2, 3, …, _b_ /_p_. This means that for every number _N_ that is not a multiple of 16, there is some _k_ such that the hex representation of _kN_ contains only the digits 1 through 8. For example, if we take the magic number at the beginning of every Java class file, 0xCAFEBABE, we find 1341 × CAFEBABEhex = 42758583546hex. In the examples above, we’re looking for multiple containing only half the possible digits. If the largest prime dividing the base is larger than 2 then we can find a multiples with digits in a smaller range. For example, in base 35 we can find a multiple containing only the digits 1 through 7. [1] Daniel Sitaru and Leonard Giugiuc. Sum of powers of the sides of a triangle. The American Mathematical Monthly, Vol. 126, No. 2 (February 2019), p. 188

Become an evolutionary detective to find the Mystery Animal!

Become an evolutionary detective to find the Mystery Animal!