Would you like to see something A M A Z I N G?

September 18, 2018 8:08 AM

Pick a number, any number and this fun little site will show how it is the sum of three palindromes! It doesn't seem like it should be possible, but it always is, in any base, and most of the time there's more than one way to do it.
posted by a snickering nuthatch (14 comments total)

The "more than one way to do it link" has this interesting observation:

The graph has a kind of self-similarity: looking at the first 100 values, there is a Gaussian-shaped peak centered at the first local maximum a(15) = 18. Looking at the first 10000 values, one sees just one Gaussian-shaped peak centered around the record and local maximum a(1453) = 766, but to both sides of this value there are smaller peaks, roughly at distances which are multiples of 10.

In the range [1..10^6], one sees a Gaussian-shaped peak centered around the record a(164445) = 57714. In the range [1..3*10^7], there is a similar peak of height ~ 4.3*10^6 at 1.65*10^7, with smaller neighbor peaks at distances which are multiples of 10^6, etc.

- M. F. Hasler, Sep 09 2018

I've added links to plots created by Hugo Pfoertner. They're worth a look.
posted by jedicus at 8:20 AM on September 18, 2018 [2 favorites]

It doesn't seem like it should be possible

counting single digit numbers as palindromes seems to make this a bit easier to believe
posted by thelonius at 8:23 AM on September 18, 2018 [14 favorites]

in any base,

Any base ≥ 5, per the linked paper. 176 (10110000₂) provides a counterxample in base 2. It remains an open question in bases 3 and 4 although the authors believe it to be true.
posted by DevilsAdvocate at 8:49 AM on September 18, 2018 [6 favorites]

Also featured on a Numberphile episode just yesterday.
posted by Hairy Lobster at 8:56 AM on September 18, 2018 [2 favorites]

Yeah, it doesn't really get interesting until you hit 6+ digits. Seeing how, for example, 8 can be represented as 8 + 0 + 0 is not compelling at all.
posted by grumpybear69 at 9:07 AM on September 18, 2018 [7 favorites]

I found the one number in the standard base that won't work, but it's my secret number only I know about.
posted by GoblinHoney at 9:24 AM on September 18, 2018 [1 favorite]

Ditto what Thelonius and Grumpy said . While I know "5" might technically be a palindrome, I don't think of it as such, because if I was sitting with friends trying to come up with palindromes, anyone who said "Oh, I've got a good one: 'E'!" would have things thrown at them.

But yes it does get cooler with huge numbers.
posted by senor biggles at 9:36 AM on September 18, 2018 [4 favorites]

I should add that it's remarkable that it can be proved for all bases >= 5! Even proving it for base 10 would be amazing
posted by thelonius at 10:19 AM on September 18, 2018

This page reminded me of zombo.com. You can do anything there, including create number palindromes.
posted by tallmiddleagedgeek at 10:19 AM on September 18, 2018 [3 favorites]

Yeah, it doesn't really get interesting until you hit 6+ digits. Seeing how, for example, 8 can be represented as 8 + 0 + 0 is not compelling at all.

Yes, but it can also be represented as 6+1+1 or 4+4+0 or 1+2+5! So fascinating.
posted by jeather at 12:06 PM on September 18, 2018

this is super neat!

also if you put in a palindrome it's just that + 0 + 0, which... i guess is right, but so unsatisfying.
posted by numaner at 2:58 PM on September 18, 2018

889 = 888 + 1 + 0

You don't say. Huh.
posted by slogger at 11:00 AM on September 19, 2018 [1 favorite]

The proof for bases 2,3 and 4, completing the theorem, was published a year after this one: Sums of Palindromes: an Approach via Automata.
posted by warpy at 6:53 AM on September 21, 2018

(And it's trivially true in unary.)
posted by a snickering nuthatch at 7:26 AM on September 21, 2018