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The following piece of music is extracted from some video on the internet (which the twitter link cannot be included here due to the content is rather violent): https://soundcloud.com/user-873168135/unknown-anime-crisis-type-music-extracted-from-anothe...
If the last digit of this POSITIVE whole number becomes the first digit, the resulting number is exactly twice as large. My number is the smallest whole number with this property. What is my number?
example: 326578 -> 832657 which clearly isn't double of 326578
You can also build 105263157894736842 one digit at a time, starting from the right, by writing …2 and then doubling the last digit you wrote and putting it next to it
Reading the question can the product of four positive integers in A.P. be a square?, also made me question whether the product of $n$ positive integers, where $n \gt 5$ in arithmetic progression be a palindrome?
Me and user Peter tried to find solutions for various $n$ in PARI/GP, and found that...
So assuming we fully understand all the individual numbers independent of all the other numbers, it's the relationships between the collective group of (positive integers) that stump us?
I mean I guess a number by itself doesn't even mean anything on its own
because there's nothing to compare it to
well maybe a number does mean something
like $5$ could represent that I have five things
but I feel like much more interesting stuff comes about when you have groups of mathematical objects and try to understand the relationships between them, not just the objects themselves
For example, I know that if you have an invertible idempotent, it's the identity, and therefore every idempotent is either a zero divisor or a multiplicative identity.
factor through? I'm not sure what that terminology means.
You can easily define a setting which forces them to be periodic. For example, just define a base system where you add a digit whenever you hit a prime. Now every prime is of the form $10^n$
@Rithaniel it means, $e \mapsto \overline{e}$ is a bijection from the idempotents in $R$ to the idempotents in $R/\mathfrak N$ where $\mathfrak N$ is the nilradical of $R$
@Rithaniel What I mean is that in some setting the primes are all right in a row, there are no numbers between them that create their disordered pattern wrt the other numbers
I wonder if there's a smaller setting to embed the integers in, such that the primes have more of a pattern
Primes in N are interesting because of their algebraic properties, so it would seem the only useful transforms would have to preserve them mostly. One possibility is to embed them in a larger setting, like the Gaussian integers. Another idea is to change the metric, as we do with p-adic numbers
Consider the following problem:
$$\nabla^2u=u+\exp(xyz)$$
in the sphere $x^2+y^2+z^2<1$ (call this $V$) and
$$\frac{\partial u}{\partial n}+u=5$$
on the surface of $S$ of the volume $V$.
Question: How do we show the solution $u(x,y,z)$ is unique or not?
My attempt: The above is in general a...
Consider the following problem:
$$\nabla^2u=u+\exp(xyz)$$
in the sphere $x^2+y^2+z^2<1$ (call this $V$) and
$$\frac{\partial u}{\partial n}+u=5$$
on the surface of $S$ of the volume $V$.
Question: How do we show the solution $u(x,y,z)$ is unique or not?
My attempt: The above is in general a...
