[seqfan] Re: n consecutive perfect powers sum to a perfect power

Tim Peters tim.peters at gmail.com
Fri Dec 15 02:28:50 CET 2023

[jnthn stdhr <jstdhr at gmail.com> wrote]

>   Would you please add a comment about a(59924) ?

No ;-) It's out of date already. I did another run through max power 10^16,
and the largest term known now is a(306700) = 9656130305203321.

I think you're going to need an a-file for this (like a b-file, but no
particular rules). For 10^16, it found 1673 solutions, but "almost all" the
fates of sequences of length less than 306700 remain unknown. A file with
1673 lines is an easy sell, but one with hundreds of thousands of
repetitions of -1 not so much.

  And, unless you care to do the tasks below for yourself, would you email
> me your list of solutions so that I can double-check my terms in the DATA
> section? I will then update the section with any new terms from your list
> and credit you.  I could also create a b-file from your list.

Certainly. Will email that next. If anyone else wants the 10^16 results,
just ask.

>   I would run your code, but I use Qpython on my phone and that uses
> python version 3.6, so it doesn't understand the  := operator.

That's shallow. If you want to play with it, just replace the body of the
relevant loop with, e.g.,

    power = i * i
    while power <= TOP:
        power *= i

But with larger max powers, I expect that, on a phone, you'll run out of
RAM. This was really written with the PyPy implementation in mind, which is
very well suited (in both speed and memory burden) to working mostly with
native 84-bit machine ints. For example, there are about 100 million powers
to chew over at max power 10^16, and under CPython just maintaining a list
of those (in a 64=bit build) burns over 4 gigabytes.

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