# Any positive number can be written as a sum of three palindromes

Originally published at: https://boingboing.net/2018/09/18/any-positive-number-can-be-wri.html

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Math!

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Here’s some background:

I’m usually not terribly excited by digit tricks, but this one interestingly can be made to work in (almost) any base.

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If you want your eyes to hurt, try verifying this in base 2.

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I love palindromes.

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You’ve a problem there. The actual theorem from the paper here says

Let g ≥ 5. Then any positive integer can be written as a sum of three base g palindromes.

I.e., they haven’t extended it to binary numbers yet.
ETA: In fact:

For g = 2 we need at least four summands. It can be checked, for example, that 10110000 is not a sum of two palindromes and it cannot be a sum of three palindromes either because it is an even number.

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My eyes already hurt from that Geocities website design.

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Are palindromic numbers important in any way?

Only on a Race Car, or A Toyota

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But wait!

What if… … … …

that design is

AMAZING!!! !?!

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I was reading it differently, since base-10 has g >= 5 any positive integer can be written as the sum of three base-10 palindromes. And that the special case is g < 5

Can’t believe nobody posted this yet:

WOW

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Let’s just post a bunch of palindromes in here. I’ll start:

Are we not pure? “No, sir!” Panama’s moody Noriega brags. “It is garbage!” Irony dooms a man—a prisoner up to new era.

Many an anemone sees an enemy anemone.

A nut for a jar of tuna.

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numbers != whole numbers

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Professional mathematician (but not number theorist) here. I’d just like to point out that the paper referenced proving this fact is 42 pages; not exceptionally long for a math paper (some of mine are shorter, some longer), but definitely long enough to make clear that the algorithm to construct the three summands isn’t at all simple. See for yourself, though - the paper’s pretty easy to follow line by line, it’s just the accumulation of steps that makes it formidable.

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Guys, this cant be true. Ive written a program to go through all positive numbers. As soon as it reaches a number that fails on this website (and it will), I’ll post the number and show this guy for the smirking fraud that he is.

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