That's Catalan's conjecture, right? So the case b = 2 is solved thanks to Mihăilescu. I don't know if that's using a sledgehammer to crack a nut or if the problem is really that hard. Charles Greathouse Analyst/Programmer Case Western Reserve University On Thu, Apr 19, 2012 at 4:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Certainly we can prove this theorem for base 2, can't we? How many cubes are there of the form 2^n-1? Something is telling me, "one," but I can't prove it.
On Thu, Apr 19, 2012 at 9:57 AM, Victor Miller <victorsmiller@gmail.com>wrote:
Charles, for the mod 14^n I get the sequence [7,37,147,502,1788,6287,22468] For mod 6^n I get [3,9,18,33,60,93,189,252,402,636]
On Thu, Apr 19, 2012 at 9:09 AM, Charles Greathouse < charles.greathouse@case.edu> wrote:
Looks about the same in those two (though of course smaller bases gain digits faster and thus have fewer examples). 6 has 1, 7, 695, and no others up to 10^7. 14 has 1, 3, 15, 21, 183, 219, 2261, 3067, 3439, 68991, 146579, 222875, 431979, 3213973, and no others up to 10^7.
In modular terms, 14 has 7, 16, 39, 95, 222, 671, 824 acceptable residues mod 14^n.
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Thu, Apr 19, 2012 at 4:11 AM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On Thursday 19 April 2012 03:57:09 Victor Miller wrote:
I'm doing some calculations pertaining to this. For all positive integers k define the set
S_k := { 1 <= n < 10^k | OddDigits(n) & OddDigits((n^3) mod 10^k) }
It looks like (log # S_k)/k --> 1 (where log is the natural log). Notice that 10^k-1 is always in S_k.
What happens in other bases? (6 and 14 seem like the closest counterparts of 10.)
-- g _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun