In any base?
Yes the theorem holds for any base.
Any positive number can be written as a sum of three palindromes
71–80 of 114 posts
Re: Any positive number can be written as a sum of three palindromes
#72For those wondering, this works in every base ≥ 5, according to the original paper: https://arxiv.org/abs/1602.06208v2 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.
> 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.
There's a stronger generalization: it's true of any divisor of the base (not just prime divisors and not just proper divisors). For example, it works for divisibility by 6 in base 12, or divisibility by 10 in base 10.
(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.)
Re: Any positive number can be written as a sum of three palindromes
#73Earlier quoted context omitted.
> 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.
There's a stronger generalization: it's true of any divisor of the base (not just prime divisors and not just proper divisors). For example, it works for divisibility by 6 in base 12, or divisibility by 10 in base 10. (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,…
And the less well known "add and subtract alternative digits" works for dividing by 11, that is, the base plus one, in any base. And for any divisors of 11.
Re: Any positive number can be written as a sum of three palindromes
#74If a(n) is the number of ways to write n as the sum of three palindromes, the sequence a(0), a(1), a(2), a(3) is kind of interesting. http://oeis.org/A261132
The graph view makes that very apparent.
Re: Any positive number can be written as a sum of three palindromes
#75Earlier quoted context omitted.
I think they're saying that 002 is counted as a palindrome, but 112 would not be. (2 == 002 but 112 != 211 which is an "unfair" property of the number zero)
If 2 is not a palindrome, because if you prepend 0 it loses its palindrome properties, then no number is a palindrome. 121 cannot be a palindrome because 121 can also be written as 0121, and therefore it’s not a palindrome. 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.
Right, that's why I put the word "unfair" in quotes; I wasn't sure how to rephrase the OP. Of course it is a real distinction, but I was trying to explain why other commenters weren't finding it impressive.
Re: Any positive number can be written as a sum of three palindromes
#76For those who are not impressed when they enter "3" and "444", try something like "8239419503490293592", and prepare to be blown away.
Re: Any positive number can be written as a sum of three palindromes
#77Wonderful wonderful website. What happends to quirky people like this when they leave university and enter the industry?
Is there such a thing as "interactive nonfiction"? Because if not, there should be. Design educational websites this way and watch children learn.
Yes! Explorable Explanations: https://explorabl.es
Re: Any positive number can be written as a sum of three palindromes
#78For those wondering, this works in every base ≥ 5, according to the original paper: https://arxiv.org/abs/1602.06208v2 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.
And according to the Numberphile video ( https://www.youtube.com/watch?v=OKhacWQ2fCs ) there is a prior paper showing it for bases 2, 3, and 4
Not base 2. All palindromes in base 2 have a 1 in the last position, because there's a 1 in the first position. This means that any addition of 3 palindromes must be odd because the result of the addition is guaranteed a 1 in the ones position, meaning even numbers can't possibly be represented that way. This is discussed in the original paper at the bottom of the second page: https://arxiv.org/pdf/1602.06208.pdf , which I've expanded out a bit since they just mention the evenness without the argument I give. They say base 2 needs at least 4 summands, but their phrasing suggests to me that's intended as a lower bound, not that they have a proof 4 is always sufficient. (Not to mention it would have to be more complicated than that anyhow since it would have to be something like "3 for odd and 4 for even above some threshold", or some other bounds.)
Re: Any positive number can be written as a sum of three palindromes
#79For those wondering, this works in every base ≥ 5, according to the original paper: https://arxiv.org/abs/1602.06208v2 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.
Yeah, for example to check if a number is divisible by 3, we can use the digit sum and check if the result is divisible by 3, e.g. 153=(1)+(5)+(3)=9.
This works in base-10, but we can generalize this by converting the dividend X into (3* X+1)-base then doing a digit sum.
Let's say we want to check whether 238 is divisible by 17, so our base is 3* 17+1=52, 238 in base-52 is (4)+(30)=34=2* 17.
If the convert-to-arbitrary-base operation was particularly cheap to compute, we could convert to base X+1 and check if the last number is 0 (same logic as: is_even = (i & 1) == 0) :)
Re: Any positive number can be written as a sum of three palindromes
#80Why three? Anyone have a theory (or the answer)?
Well, there is a whole 40+ page paper on the algorithm and its proof of correctness. So, yes there is an answer. https://arxiv.org/abs/1602.06208