Any positive number can be written as a sum of three palindromes
somethingorotherwhatever.com
somethingorotherwhatever.com
Every time I see something like this that is dependent on base-10 representation, I'm always curious to know whether it generalizes, or if it's a quirk specific to base-10 representation. For example, being able to tell if a number is divisible by 2 or 5 by only looking at the last digit.
That still seems like it is generalizable to any base by taking prime factors of any base-n.
(Also, the digital root test, where you add up the digits, works for any divisor that is also a divisor of the base minus 1, so for example hexadecimal has a digital root test for divisibility by 3, 5, or 15.)
2 is (2,0,0) and (1,1,0).
A composition is an ordered partition. A partition is the set of numbers that sum to a particular number. The 'weak' part allows zeros.
Without thinking the problem through (about to sleep) I wouldn't be surprised if the answer is easily proven to be exponential (aka a firm "nope") but math can be unintuitive at times so I thought I'd ask anyway.
I ask because, if the maximum number of potential matches isn't exponential (or is at least <= 3 digits long), it might be plausible to read through the number of matches to find "interesting" (elegant-looking) ones.
What happends to quirky people like this when they leave university and enter the industry?
But as a one-shot, I love it!
The real physical Universe is interactive nonfiction.
The real physical universe is nothing but tooling for playing with advanced math (but it's not always easy to figure out what the advanced math is, or how best to play with it).
And not just the learning purposes. Imagine the possibilities if we could properly simulate (and speed up the simulation) the world - what if we could use genetic algorithms to develop an organism.
These two statements are quite different. Higher level programming languages than assembly are unnecessary; anything beyond the lambda calculus is unnecessary (assuming our goal is to compute Turing computable things). That's very different from saying that they're useless; just because they can be done without doesn't mean that they should.
I wish there was a list of sites like these - learning for the pure joy of learning, with no other expectations.
Newcastle University: I'm e-learning officer in the School of Mathematics and Statistics.
Most of my job involves writing the maths e-assessment system, Numbas. https://www.numbas.org.uk/
For those wondering how to generate random numbers, it's simple if you don't care about a little computational overhead: </dev/urandom tr -dc 0-9 | head -c 15 (substitute 0-9 for any character set like a-zA-Z or \ -~). I use this to generate random PIN numbers after it took too long to read the pwgen man page, and I might start to use it as replacement for pwgen altogether since it only uses very basic tools that are available everywhere, unlike pwgen.
To explain the command: we read urandom into tr's stdin; tr translates (like "echo haha | tr a u"); with -d, tr deletes instead of substituting; with -c, it takes the compliment set (so everything except what you specify). Basically we're filtering characters or of urandom's output. Finally, we limit the output to a certain number of characters.
It's a cool result.
LC_CTYPE=C </dev/urandom tr -dc 0-9 | head -c 15
Do you just mean "the sequence a(n)"? I don't see anything especially interesting about the first 4 terms.
(2 == 002 but 112 != 211 which is an "unfair" property of the number zero)
Further, there is nothing “unfair” about 0. It’s a real distinction. 002 is the same thing as 2. 112 is not the same thing as 2. So it’s not an unfair property. It’s a real distinction.
101=101+0+0
> Yes, I can do this for any whole number bigger than zero.
Should say "Yes, I can do this for any whole number zero or bigger." since 0 is a palindrome
Where an arbitrarily complex time-series signal can be accurately represented as the sum of various sine waves.
Except the OP deals with a discretized values, not real-number values, and the OP's approach lets the constituent palindromes start/end at an arbitrary offset, rather than having to be repeated continuously.
If the number itself happens to be a palindrome, you can get down to 50%, but that's completely unrelated to this trick.
In general, it's impossible for a compression algorithm to losslessly compress all numbers to a shorter representation, it can only make some numbers shorter and others longer. (Pigeonhole principle: there are not enough short codes, so if you assign a shorter code to all numbers, some will get the same code, making the compression not lossless.)
Practical uses of compression are all about reducing the average length given a nonuniform probability distribution, so a palindrome-based compression scheme would only be useful if you could expect most numbers you're dealing with to be palindromes.
in all seriousness it is always amazing to see simple theorems rely on such complex proofs
77777777777777 +0 +0 =77777777777777
> 1,234,567 + 0 + 0 = 1,234,567
The first of these is not a palindrome.
:)
And if you don't like that:
55555 = 12321 + 23132 + 20102
Palindromes are sort of the least surprising numbers to have this property because you can easily "split" them into as many palindromes as you want like this. The surprising result is that you could write something like 19837100018374 as the sum of only 3 palindromes.