The base 2 list is even shorter.
Powers of 2 with all even digits
11–20 of 123 posts
Re: Powers of 2 with all even digits
#12Somehow I missed the title and wondered what the fuck was going on... 2, 4, 8, 64, 2048 are powers of 2 (i.e. 2^n), and they don't contain odd numbers (e.g. 16, 128, 1024 contain 1 so are not in this list, same with 4096 containing 9).
I'm confused by your comment. First, powers of two are 2^n not n^2. But what do you mean you missed the title and wondered what was going on? How could you expect to understand the contents without reading the title? Surely I'm missing something.
Re: Powers of 2 with all even digits
#13No additional terms up to 2^(10^10). - Michael S. Branicky, Apr 16 2023 How did he do this?
As noted in 3) in the Shepherd's comment, 2^k has no odd digits when 2^k mod 10^n for all integer n have no odd digits as well. So many k would be filtered by checking whether 2^k mod 100 has an odd digit, then another portion of the remainder will get filtered with 2^k mod 1000, 2^k mod 10000 and so on. (EDITED: Thanks to andrewla!) All of them would be periodic, so first few steps can be made into a lookup table to…
> whether 2^k mod 10 is odd
2^k mod 10 is never odd; it's the cycle (2, 4, 8, 6).
Related here is the length of the cycles mod 2^k, https://oeis.org/A005054. Interestingly, the number of all-even-digit elements in those cycles does not appear to be in the oeis, I get 4, 10, 25, 60, 150 as the first five terms.
This does appear to get more efficient as k gets higher; for k=11 I get a cycle length of 39,062,500 with an even subset of 36,105, meaning only .09% of the cycle is all-even.
This is all brute force; there's probably a more elegant way of computing this.
Re: Powers of 2 with all even digits
#14[flagged]
Re: Powers of 2 with all even digits
#15[flagged]
This is about 2^n, not n^2. 2^0 is 1, which does not fit "all digits are even".
Re: Powers of 2 with all even digits
#16Earlier quoted context omitted.
As noted in 3) in the Shepherd's comment, 2^k has no odd digits when 2^k mod 10^n for all integer n have no odd digits as well. So many k would be filtered by checking whether 2^k mod 100 has an odd digit, then another portion of the remainder will get filtered with 2^k mod 1000, 2^k mod 10000 and so on. (EDITED: Thanks to andrewla!) All of them would be periodic, so first few steps can be made into a lookup table to…
> whether 2^k mod 10 is odd 2^k mod 10 is never odd; it's the cycle (2, 4, 8, 6). Related here is the length of the cycles mod 2^k, https://oeis.org/A005054 . Interestingly, the number of all-even-digit elements in those cycles does not appear to be in the oeis, I get 4, 10, 25, 60, 150 as the first five terms. This does appear to get more efficient as k gets higher; for k=11 I get a cycle length of 39,062,500 with a…
Oh, yeah, I should have said 2^k mod 100 has no odd digits.
Re: Powers of 2 with all even digits
#17[flagged]
The link is about 2^n not n^2.
Re: Powers of 2 with all even digits
#18Re: Powers of 2 with all even digits
#19Earlier quoted context omitted.
As noted in 3) in the Shepherd's comment, 2^k has no odd digits when 2^k mod 10^n for all integer n have no odd digits as well. So many k would be filtered by checking whether 2^k mod 100 has an odd digit, then another portion of the remainder will get filtered with 2^k mod 1000, 2^k mod 10000 and so on. (EDITED: Thanks to andrewla!) All of them would be periodic, so first few steps can be made into a lookup table to…
> whether 2^k mod 10 is odd 2^k mod 10 is never odd; it's the cycle (2, 4, 8, 6). Related here is the length of the cycles mod 2^k, https://oeis.org/A005054 . Interestingly, the number of all-even-digit elements in those cycles does not appear to be in the oeis, I get 4, 10, 25, 60, 150 as the first five terms. This does appear to get more efficient as k gets higher; for k=11 I get a cycle length of 39,062,500 with a…
10^10 * 36105/39062500 = 9242880, so you're already down to under 10^7 cases to check, which is starting to seem more tractable.
Re: Powers of 2 with all even digits
#20This is remarkable! I always find it fascinating that simple to express properties lack a proof. This is a very simple thing to evaluate and seems like it should be straightforward to establish that 2048 is the highest such power.
Proofs of non-existence aren't usually straightforward.