Simplification of "reciprocal word 3by2" (aka invert_pi1)

Paweł Bylica chfast at
Sat Jun 20 08:26:11 UTC 2020

Hello Paul,

I confirm this is a valid counter-example. Thank you very much.

If that is not a problem, are you able also to generate a case for t0 == d0?

// Paweł

On Sat, Jun 20, 2020 at 7:53 AM Paul Zimmermann <Paul.Zimmermann at>

>        Dear Pawel,
> > Thanks for checking this out Paul.
> >
> > Are you able to generate any counter-example for beta=2^64?
> yes: d1=9223372036855824384 d0=9223374235880128513.
> Please can you confirm? This should be the smallest solution (by
> d1*2^64+d0).
> How did I? I wrote a SageMath program that for each d1 computes the ranges
> of d0 values that correspond to the same final value of v after line 9.
> Then I kept ranges of d0 that give p < t1 at line 12, and for which
> (beta+v-1)*(d1*(v-1)+d0) >= beta^3, which is the condition for the second
> adjustment at line 15. It then remains to check the smallest d0 of each
> range.
> Paul

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