Intereting Posts

A dilogarithm identity?
$\mathbb{F}_p/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
Polar decomposition of real matrices
Number of bitstrings with $000$ as substring
Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.
When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated?
Prove that the convex hull of a set is the smallest convex set containing that set
probability distribution of coverage of a set after `X` independently, randomly selected members of the set
Uniform convergence in distribution
Give an equational proof $ \vdash (\forall x)(A \rightarrow B) \equiv ((\exists x) A) \rightarrow B$
Existence of fixed point given a mapping
Given 5 children and 8 adults, how many ways can they be seated so that there are no two children sitting next to each other.
Geometric intuition for the tensor product of vector spaces
On the definition of free products
Galois group and the Quaternion group

The number $4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number.

$3883544142410_{10}=3883544E24A_{16}$ is another.

These numbers are in OEIS A187829. They come in blocks of $6$ or $10$, depending on whether the one’s digit in hex is $A-F$ or $0-9$.

- Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$
- Factor $14x^2 - 17x + 5$.
- General misconception about $\sqrt x$
- How find the value of the $x+y$
- how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
- Three-variable system of simultaneous equations

I suspect the list is complete but have not proven it. The largest is $806123145829415507126939101294137128298625241370656314360169_{10}=\\806C3E58294F507C6939AC94D7C829862524D706563E360169_{16}$

If the number has $m$ hex digits and $n$ base $10$ digits, we must have $16^{m-1} \gt 10^{n-1}$ and $16^{m-2} \lt 10^{n-2}$ which leads the hunt to $m=6,n=7;\ m=11,n=13;\ m=16, n=19;\ m=50,n=60$ guided by the convergents of $\frac {\log 16}{\log 10}$.

We can view finding these numbers as finding solutions to the subset-sum problem, where each hex digit contributes the difference between its value in base $16$ and base $10$ (depending on how many base $16$ digits to the right are $A-F$ and counting the two base $10$ digits coming from one hex digit together). The sum then has to be zero.

My search program ran reasonably quickly even for the next convergent, $m=535, n=644$ and didn’t find any. I believe they just have too many ways to fail as the number gets long.

Can we prove that there are no more, or at least that there are no more with very high probability, in the sense of “proofs” of Goldbach that if the primes are “random” the chance of any large even number having no solution is very low?

- How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$
- How to algebraically prove $\binom{n+m}{2} = nm + \binom{n}{2} + \binom{m}{2}$?
- What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?
- About finding the period of a function which satisfies $f(x-2)+f(x+2)=f(x)$.
- Simple algebra question - proving $a^2+b^2 \geqslant 2ab$
- Don't understand casting out nines
- Confusion regarding taking the square root given an absolute value condition.
- Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in $
- Beautiful cyclic inequality
- what operation repeated $n$ times results in the addition operator?

I can prove a very narrow form.

Let’s consider numbers of the form:

$(D_{n-1}D_{n-2}…..D_{2}D_{1}D_0)_{10} = (D_{n-1}D_{n-2}…..D_2[A..F])_{16}$.

Here 2 least significant digits in decimal representation change

to [A..F]. For these numbers, conditions that need to satisfy are,

$100*x + 10 = n$ ….(1)

$16*y + 10 = n$ ….(2)

So,

$y = 6.25*x$ ….(3)

Let’s assume,

$x_{10}$ is of form $….N_{k-1}N_{k-2}…..N_{2}N_{1}N_{0}$

or,

$x_{10} = ….+ (N_{k-1}10^{k-1}) + (N_{k-2}10^{k-2}) + ….+ (N_{2}10^2) + (N_{1}10^1) + N_{0}$

So,

$y_{10} = ….+ (N_{k-1}16^{k-1}) + (N_{k-2}16^{k-2}) + ….+ (N_{2}16^2) + (N_{1}16^1) + N_{0}$

Also from (3),

$y_{10} = 6.25(….+ (N_{k-1}10^{k-1}) + (N_{k-2}10^{k-2}) + ….+ (N_{2}10^2) + (N_{1}10^1) + N_{0})$

So,

$(….+ (N_{k-1}16^{k-1}) + (N_{k-2}16^{k-2}) + ….+ (N_{2}16^2)

+ (N_{1}16) + N_{0}) =

6.25(….+ (N_{k-1}10^{k-1}) + (N_{k-2}10^{k-2}) + ….+ (N_{2}10^2)

+ (N_{1}10^1) + N_{0})$

Using up to 6 digits for x,

$(1048576N_5 + 65536N_4 + 4096N_3 + 256N_2 + 16N_1 + N_0) =

(625000N_5 + 62500N_4 + 6250N_3 + 625N_2 + 62.5N_1 + 6.25N_0)$

or,

$42357600N_5 + 303600N_4 – 215400N_3 – 36900N_2 – 4650N_1 – 525N_0 = 0$

Value in Hex falls behind till N3 because initial modulus was 100 for decimal

and 16 for hex. But at and after N4, hex value overtakes decimal forever.

Only solution for N5 (and above) between 0 and 9 is 0. Also, there are no solutions possible less

than 5 digits for x in (1).

So essentially numbers of these forms are only 7-digits or 2 digits,

And only possible solutions are,

```
10-15
4494410-4494415
5660810-5660815
6784010-6784015
7950410-7950415
```

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- Getting the grip of geometry and Algebra; books and resources for a beginner
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- Set theory practice problems?
- Simpler way to compute a definite integral without resorting to partial fractions?
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- First Pontryagin class on real Grassmannian manifold?
- Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$
- Sylow questions on $GL_2(\mathbb F_3)$.
- Subgroups of Prime Power Index
- Graph Theory Path Problem
- Irreducible polynomials and affine variety