Re: Power anagrams

[ Mark Brader wrote: ]
Find instances where two consecutive nontrivial powers of the same
base (i.e. p^n and p^(n+1) for integers p,n > 1) are anagrams of two
consecutive nontrivial powers of a different base, not necessarily
respectively. For example, 2^10 = 1024 and 7^4 = 2401, and 2^11 = 2048;
if either 7^3 or 7^5 was 8402, that would be a solution.

I did a computer search of numbers of up to 12 digits and found 26
solutions, but most of them involved numbers with multiple zeroes
in them, formed in essentially trivial ways. It's not very interesting
or even surprising that 1001^2 and 1001^3, and 1010^2 and 1010^3,
are respectively anagrams of each other (and of 1100^2 and 1100^3
too, for that matter).

The two smallest cases, however, involve no zeroes. I reckon you'll
need computer assistance to find one, but not the other. Well?

And you were right, at least for me. The solutions have already been
posted by others: 5^{3,4} =~ 2^{9,8} and 178^{2,3} =~ 196^{2,3}. I
thought I would note, however, that one of the other solutions in that
range with zeroes appears to be non-trivially interesting:

3230^2 = 10432900
3302^2 = 10903204
3230^3 = 33698267000
3302^3 = 36002379608

[ And these bases are anagrams of each other, so we actually have
3230^{1,2,3} =~ 3302^{1,2,3}. ]

Extending the search up to 15 digits produces the following non-
trivial
example with an exponent of interest:

1851^{3,4} =~ 43641^{2,3}

The rest of the non-trivial additions found all use exponents 2 and 3,
so I shall omit them and just show the base pairs:

12036, 14016
25565, 30875
25727, 29291
46713, 47763
50092, 66355
60332, 99815
83902, 95116
89637, 97356
94272, 94767

Extending the search up to 18 digits yields a large number of other
solutions. One of these also uses no zeroes:

546141^2 = 298269991881
996141^2 = 992296891881
546141^3 = 162897471635881221
996141^3 = 988467618175231221

Cheers,
Geoff.

-----------------------------------------------------------------------------
Geoff Bailey (Fred the Wonder Worm) | Programmer by trade --
ftww@xxxxxxxxxxxxxxxxx | Gameplayer by vocation.
-----------------------------------------------------------------------------

.

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