What is the base of the next nice number?
closes 2031
49 - 85
86 - 100
101 - 110
111 - 120
121 - 125
126 - 130
131 - 150
There are no more nice numbers

A nice number is a natural number such that its square and cube combined contain all digits exactly once, without needing to use any leading zeros. It is already known that 69 is the only nice number in base ten, and so far, no nice number has been found in any other base either.

Currently, the bases up to 48 have been completely searched for nice numbers, with none found, aside from 69 in base 10. You can see https://nicenumbers.net/ for the current status of the serach, as well as @Conflux's last post about it: https://beautifulthorns.wixsite.com/home/post/progress-update-on-the-search-for-nice-numbers

As you can see, it is estimated that there should be about one nice number by the time we get to base 118, but this is only an estimate.

This market resolves once the next nice number is found, provided that all bases smaller than the one it is nice in have been exhaustively checked (or otherwise proven not to have nice numbers). If it is proven that 69 is the only nice number, this will resolve to the last option. If it turns out that somehow the code for the seraches that have been done so far was wrong and there actually is another nice number in a base below 49, then I'll resolve to the "49 - 85" option.

Get Ṁ500 play money

Related questions

6 ÷ 2(1+2) =
Cover's paradox: Is 29 the larger of my two secret numbers?
JosephNoonan avatarPlasma Ballin'
42% chance
What is the smallest range that contains the second number of Plasma Ballin's Cover's paradox market?
Is 0 a natural number?
ithildulin avatarAmelia
66% chance
Is the answer to the Sleeping Beauty Problem 1/3?
MartinRandall avatarMartin Randall
55% chance
Is 5 the only semi-nice number?
JosephNoonan avatarPlasma Ballin'
10% chance
Number of Watts Needed to Run The First Implementation of AGI?
Algebraic value editing works better for larger language models, all else equal
MartinRandall avatarMartin Randall
51% chance
Even or odd number of transactions?
HenriqueJesus avatarHenrique Jesus
43% chance
Is 69 the only nice number?
In 2020, Joe Carlsmith estimated that 10^15 FLOPS is "enough" to perform human functionality. The next post estimating this number will believe it is >=10^15
Quinn avatarQuinn Dougherty
36% chance
HannahFox avatarHannah Fox
63% chance
What is your intuition about this probability comparison?
1️⃣0️⃣0️⃣0️⃣0️⃣0️⃣0️⃣0️⃣0️⃣0️⃣➕ 💩🥄😋?
What is the last decimal digit of 2^(3^(4^(5^6)))
jacksonpolack avatarjackson polack
Is the 20th Busy Beaver number independent of set theory?
BoltonBailey avatarBolton Bailey
31% chance
Is 10 a friendly number?
JosephNoonan avatarPlasma Ballin'
27% chance
Is 196 the first *true* Lychrel number?
JosephNoonan avatarPlasma Ballin'
67% chance
Binary Induction Game v1
Consider the original Monty Hall problem and a modified version with 1000 doors. Which one of the following applies?
Sort by:
FlorisvanDoorn avatar
Floris van Doornbought Ṁ10 of 121 - 125 YES

Just saw these markets for the first time. I don't have much time to spend on this now, but I got nerd-sniped, and I might have some observations.

  • I quickly skimmed the "Progress Update on the Search for Nice Numbers" post, and some parts seem incredible sketchy (wrong?) to me. Are you claiming that for bases 3 mod 4 there are no nice numbers? I don't believe that. Are you sure the argument is not accidentally using the (wrong) claim that digit-sum(x^2)=(digit-sum(x))^2 mod (b-1)?

  • Is there any exponential improvement over brute force search for nice numbers? Or is the complexity to check all cases for base b still roughly O(b^(b/5))? I have some ideas to do this exponentially faster (probably still not fast enough to get to another nice number)

4 replies
FlorisvanDoorn avatar
Floris van Doorn

@FlorisvanDoorn oh ignore my first point. The argument seems valid. I should stop thinking about this and go to sleep.

JosephNoonan avatar
Plasma Ballin'bought Ṁ1 of 126 - 130 YES

@FlorisvanDoorn The proof in the blog post goes through some steps quickly, but it is definitely valid. To spell it out fully, if x is nice, then x^2+x^3 ≡ b(b-1)/2 (mod b-1). This is because any number is congruent to the sum of its digits in base b, mod b-1, and we know that the sum of the digits in x^2 and x^3 must be the (b-1)th triangular number, b(b-1)/2. If b ≡ 3 (mod 4), then b must be odd, so (b-1)/2 is an integer. This means that b(b-1)/2 ≡ (b-1)/2 (mod b-1), since b ≡ 1 (mod b-1). Then, writing b as 4k+3, we find that x^2+x^3 ≡ 2k+1 (mod b-1) for some k. But 2k+1 is odd, while b-1 is even, so this tells us that x^2+x^3 is odd, since it's residue modulo an even number is odd. On the other hand, we know that x^2+x^3 is always even for any integer x. This is a contradiction, so there are guaranteed to never be nice numbers for bases that are 3 mod 4.

JosephNoonan avatar
Plasma Ballin'

@FlorisvanDoorn I just finished writing the proof, and only then did I see your second comment

JosephNoonan avatar
Plasma Ballin'

@FlorisvanDoorn As for the first one, yeah, I'm pretty sure no one has found an exponential improvement over brute force.

Conflux avatar
Confluxbought Ṁ5 of 101 - 110 YES

Just remembered this market should have a pretty large skew toward lower bases since it’s more likely than a nice number there will be discovered