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Re: Re: Re: Re: smallest

• To: mathgroup at smc.vnet.net
• Subject: [mg86869] Re: [mg86833] Re: [mg86828] Re: [mg86792] Re: [mg86771] smallest
• From: danl at wolfram.com
• Date: Mon, 24 Mar 2008 01:45:05 -0500 (EST)
• References: <200803200757.CAA29500@smc.vnet.net>

> If we want to find rational fraction f =p/q such that 113/355<f<106/333
> and sum p+q is minimal
> anyone procedure proposed  up to now doesn't work
> good result should be
> {137563,{p->13215,q->104348}}
> but isn't
> ARTUR

You can use continued fractions to get a contender, as below.

In[8]:= ContinuedFraction[355/113]
Out[8]= {3, 7, 16}

In[15]:= ContinuedFraction[333/106]
Out[15]= {3, 7, 15}

What this suggest is that a plausible result might be obtained by
splitting the difference on the last convergent:

1/(3+
     1/(7+
          2/31)))

This yields 219/688.

That happens to agree with Minimize:

In[17]:= Minimize[{p+q, {355*p>113*q,333*p<106*q,p>=1,q>=1}}, {p,q},
Integers]

Out[17]= {907, {p -> 219, q -> 688}}

I notice that 907 is considerably smaller than 137563.

Daniel Lichtblau
Wolfram Research

• References:
  • smallest fraction
    • From: masmoudi <mas_atef@yahoo.fr>