Find three distinct positive integers p, q and r such that the value of expression 1/p+1/q+1/r will be less than 1/2 but as close to it as possible.

Find four distinct positive integers p, q, r and t such that the value of expression 1/p+1/q+1/r+1/t will be less than 1 but as close to it as possible.

   10   for P=3 to 9999
   20     Rp=1//P
   30     if 3*Rp<Max then cancel for:goto *Pend
   40     for Q=P+1 to 9999
   50       Rq=1//Q
   60       if Rp+2*Rq<Max then cancel for:goto *Qend
   70       for R=Q+1 to 9999
   80         Rr=1//R
   90         Tot=Rp+Rq+Rr
  100         if Tot<Max then cancel for:goto *Rend
  110         if Tot<1//2 and (Tot<Max or Max=0) then
  120           :Max=Tot:Mp=P:Mq=Q:Mr=R
  130        next R
  140      *Rend
  150      next Q
  160    *Qend
  170    next P
  180  
  190    *Pend
  200    print Mp;Mq;Mr,Max,Max/1
  205    print
  210    Max=0
  220    for P=2 to 99999
  230      Rp=1//P
  240      if 4*Rp<Max then cancel for:goto *Pend1
  250      for Q=P+1 to 99999
  260        Rq=1//Q
  270        if Rp+3*Rq<Max then cancel for:goto *Qend1
  280        for R=Q+1 to 99999
  290          Rr=1//R
  300          if Rp+Rq+2*Rr<Max then cancel for:goto *Rend1
  310          for T=R+1 to 99999
  320            Rt=R//T
  330            Tot=Rp+Rq+Rr+Rt
  340            if Tot<Max then cancel for:goto *Tend1
  350            if Tot<1 and (Tot<Max or Max=0) then
  360              :Max=Tot:Mp=P:Mq=Q:Mr=R:Mt=T
  370           next T
  375         *Tend1
  380         next R
  390       *Rend1
  400       next Q
  410     *Qend1
  420     next P
  430   
  440     *Pend1
  450     print Mp;Mq;Mr;Mt,Max,Max/1

finds

 p  q  r   t               total
 3  7  43        451/903        0.4994462901439645625

 2  3  7  295    12389/12390    0.9999192897497982243

Edited on November 25, 2012, 12:43 pm

Posted by Charlie on 2012-11-25 12:40:34