[seqfan] Re: 32/25 - a Northern Summer puzzle

[Peter Munn]

On Wed, June 29, 2022 1:34 am, wnmyers wnmyers wrote:
> Clearly the asymptote must be more than 5 because you have to divide by 5
> at some point and get a number greater than one. The 6-step route gives an
> upper bound of 6.4, so we can ignore any multiplication or division by 7
> or more. That means all bounds after that would be of the form
> 2^a*3^b*5^c.
>
> steps  bound
> 4      8=2^3
> 6      6.4=2^5*5^-1
> 12     5.689=2^8*3^-2*5^-1
> 26     5.619=2^12*3^-6
> 76     5.605=2^-12*3^15*5^-4
>
> I'm pretty sure the bound can't go any lower than 5.346=2^28*3^-22*5^4.

That all looks good, it conforms with my analysis.

Unless my analysis is mistaken, for all 5-smooth rationals, q, 1 <= q <=
U_min, where U_min is the asymptote from my puzzle: either (1) the
asymptote for q is U_min, or (2) there is a lowest upper bound U_q <=
U_min which permits q to be reached (therefore with a finite number of
steps). If q is above U_min there is a route to q that does not exceed q.

As a hint, the product of the numerator and denominator of my value for
U_min is less than 1000.

Peter