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Powers of 2 with all even digits

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Re: Powers of 2 with all even digits

#41

Post by Sharlin »

LegionMammal978 wrote: post #39
Sharlin wrote: post #33

Earlier quoted context omitted.

True. It might also just be that the question hasn't attracted the attention of number theorists, and finding a proof wouldn't be unreasonably difficult to an expert in the field.

Nope, it's not that easy in this case. E.g., Erdős conjectured in 1979 that every power of 2 greater than 256 has a digit '2' in its ternary expansion [0]. This makes sense heuristically, but no methods since then have come close to proving it. Digits of numbers are a wild beast, and they're tough to pin down for a specific sequence. At best, we get statistical results like "almost all sequences of this form have thi…

Thanks!

Re: Powers of 2 with all even digits

#42

Post by vhcr »

38 wrote: post #4
WithinReason wrote: post #3

No additional terms up to 2^(10^10). - Michael S. Branicky, Apr 16 2023 How did he do this?

yeah that's weird - its kind of a pointless comment without an included algorithm or something

Here's a really dumb algorithm:

    for i in range(1, 10**10):
        for k in range(1, 5):
            s = str(pow(2, i, 10**(10**k)))
            if '1' in s or '3' in s or '5' in s or '7' in s or '9' in s:
                break
        else:
            print(2**i)
It's really easily to parallelize, I was able to run it up to 10**8 in about 15min, so you would be able to run it up to 10**10 in a few hours with parallelization.

Re: Powers of 2 with all even digits

#43

Post by LegionMammal978 »

codeflo wrote: post #34
andrewla wrote: post #7

This 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.

Everything about this seems so arbitrary. You look at the powers of an arbitrary number (here, 2), you pick an arbitrary base (here, 10) in which to express those powers, and ask for a random property of its digits (whether they belong to the set {0,2,4,6,8}). Nothing about this question feels natural. I've noticed that random facts often don't have simple proofs.

In this case, it doesn't even help to downsize the problem. Erdős once asked the same question, but with powers of 2, base 3, and the set {0,1}. (If you want to, you can disguise that version as something more natural-looking like "Which powers of 2 can be expressed as the sum of distinct powers of 3?") But we're still nowhere close to solving it.

Re: Powers of 2 with all even digits

#44

Post by 1970-01-01 »

nickvec wrote: post #36
1970-01-01 wrote: post #32

Earlier quoted context omitted.

I see the difference in wording now, as its not very clear what they meant

Saying “powers of two” is a universal way of denoting 2^n. It’s okay to admit being wrong rather than blaming OEIS for being vague.

The phrase squared is "powers of two" is much, much more common than you know. I'm part-time substituting as a grade-school teacher. Ask any if they are explaining this difference.

Re: Powers of 2 with all even digits

#46

Post by ABS »

1970-01-01 wrote: post #44
nickvec wrote: post #36

Earlier quoted context omitted.

Saying “powers of two” is a universal way of denoting 2^n. It’s okay to admit being wrong rather than blaming OEIS for being vague.

The phrase squared is "powers of two" is much, much more common than you know. I'm part-time substituting as a grade-school teacher. Ask any if they are explaining this difference.

"power of two" with "power" singular perhaps but "powers of two" with the word "powers" plural is really not common as a synonym of squared (excluding people who don't know what they mean at all in the first place)

Re: Powers of 2 with all even digits

#47

Post by madcaptenor »

lanna wrote: post #30

How many powers of 2 have just a single even digit? 2, 4, 8, 16, 32, 512...

Looks like that's all of them. The typical number of even digits of n grows like a constant times n, so you need some very large deviations from t

I'd conjecture the number of powers of 2 with exactly m even digits is finite for all m.

Re: Powers of 2 with all even digits

#48

Post by toxik »

vhcr wrote: post #42
38 wrote: post #4

Earlier quoted context omitted.

yeah that's weird - its kind of a pointless comment without an included algorithm or something

Here's a really dumb algorithm: for i in range(1, 10**10): for k in range(1, 5): s = str(pow(2, i, 10**(10**k))) if '1' in s or '3' in s or '5' in s or '7' in s or '9' in s: break else: print(2**i) It's really easily to parallelize, I was able to run it up to 10**8 in about 15min, so you would be able to run it up to 10**10 in a few hours with parallelization.

It's not 10^10 ≈ 2^33 though, it's 2^(10^10) = 2^10000000000, or about 9 999 999 967 orders of magnitude more.

Re: Powers of 2 with all even digits

#49

Post by vessenes »

IsTom wrote: post #45

It might be finite, but it also has a "fast growing sequence" kind of smell too.

I thought that at first as well. Then I read the notes which made me reframe it as ‘odds your digit sequence won’t include a six ever’ and note that checking up to 2^50000 has only two candidates with the first 15 digits even, and I came down on ‘shrinking so quickly it’s super unlikely’. No proof here due to HNs comment limits of course..

Re: Powers of 2 with all even digits

#50

Post by showerst »

toxik wrote: post #48
vhcr wrote: post #42

Earlier quoted context omitted.

Here's a really dumb algorithm: for i in range(1, 10**10): for k in range(1, 5): s = str(pow(2, i, 10**(10**k))) if '1' in s or '3' in s or '5' in s or '7' in s or '9' in s: break else: print(2**i) It's really easily to parallelize, I was able to run it up to 10**8 in about 15min, so you would be able to run it up to 10**10 in a few hours with parallelization.

It's not 10^10 ≈ 2^33 though, it's 2^(10^10) = 2^10000000000, or about 9 999 999 967 orders of magnitude more.

Just by sheer numbers, the comment you're replying to must be one of the provably wrong-est comments in history of hacker news =).
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