[seqfan] Re: Numbers not sums of perfect powers

Benoît Jubin benoit.jubin at gmail.com
Thu Apr 15 05:25:17 CEST 2010


On Wed, Apr 14, 2010 at 7:31 PM,  <franktaw at netscape.net> wrote:
> 2^k and 3^k are relatively prime. The set of numbers that cannot be
> expressed as a sum using any set of relatively prime numbers is finite.

Yes, this was in my original post:
---
For analogues of the third sequence, we have:
The numbers which are not the sum of k^th powers larger than one are
exactly those in [1,6^k-3^k-2^k] but not of the form
2^k.a+3^k.b+5^k.c.  This relies on the following fact: if m and n are
relatively prime, then the largest number which is not a linear
combination of m and n with positive integer coefficients is mn-m-n.

Is there an analogous result for relatively prime integers {m_i} ?
---

You answered positively to my last question: do you know a simple
expression for the largest number which cannot be written in this way?
 For the numbers which are not sums of kth-powers-minus-1, we have to
show that the latter are relatively prime, and it would be even better
to find a effective bound...

Thanks,
Benoit


>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Benoît Jubin <benoit.jubin at gmail.com>
>
> ..  And I'm still interested in
> finiteness for the general case (k^th powers).
>
>
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