I suspect that this is one of those cases where we could produce a heuristic argument that strongly suggests that the "probability" of finding a larger looper is very small -- and we could make it smaller by extending the search higher. But I don't see any promising proof techniques. On Wed, Dec 30, 2020 at 12:17 AM W. Edwin Clark <wclark@mail.usf.edu> wrote:
Allan, you are right! It occurred to me after I pressed the send button that I was missing a proof. I only show that those I listed are the only reduced loopers up to 10^6. In particular there are no reduced loopers with 4, 5 or 6 digits. There could be one with 7 or more digits. --Edwin
On Wed, Dec 30, 2020 at 12:01 AM Allan Wechsler <acwacw@gmail.com> wrote:
I think one of Éric's examples made it clear that the procedure abducts the rightmost occurrence of the smallest digit and turns it into an exponent.
I am curious to know how Edwin proved the completeness of his reduced looper set. I mean, how do you know there isn't some biggish number that gets squared or cubed a few times, until it's really gigantic, and then happens to be made mostly of 1's, which when removed leave the original biggish number?
On Tue, Dec 29, 2020 at 11:49 PM Dan Asimov <asimov@msri.org> wrote:
Eric Angélini a écrit:
Starting from the right, pick D’s smallest digit
What does *where you start* have to do with this?
And what is the rule for when there is a tie?
(52792, for instance.)
—Dan
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