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Curious Cubic Constant (Posted on 2009-05-02) Difficulty: 3 of 5
Determine the minimum value of a positive integer constant c such that the equation xy3 - y3+ x + y = c has precisely four distinct solutions in positive integers.

See The Solution Submitted by K Sengupta    
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Solution Smallest c for 1,2,3,4,5,6 distinct solutions Comment 3 of 3 |
One solution is trivial with c=2: (1,1)
Two solutions is also trivial with c=4: (3,1) and (1,2)
Three solutions is easy to find with c=12: (6,1), (2,2), and (1,11)

Four solutions is where things started getting nontrivial.  I started by rearranging the equation into x = [c/(y+1) - 1] / [y^2-y+1] + 1.  This means that y+1 must be a factor of c.  Also the only possible solution with y>cbrt(c) is (1, c-1).  These two facts greatly reduces the search space for potential solutions.

The Ubasic program below searches for solutions:
   10   Cmin=8
   11   Cmax=10000
   20   for C=Cmin to Cmax
   22   Count=0
   25   Ymax=floor(C^(1/3))
   30   for Y=1 to Ymax
   40   F=C(Y+1)
   41   R=res
   50   if R>0 then 100
   60   D=Y*(Y-1)+1
   70   X=((F-1)D)+1
   71   R=res
   80   if R>0 then 100
   90   ' print X,Y,C
   91   Count=Count+1
   92   if Count<3 then 100
   95   print C,Count
  100   next Y
  110   next C

Running this program finds the smallest c for 4, 5 and 6 solutions:
Four solutions with c=200: (100,1), (8,3), (4,4), (1,199)
Five solutions with c=4008: (2004,1), (446,2), (144,3), (4,11), (1,4007)
Six solutions with c=274404: (137202,1), (30490,2), (9801,3), (207,11), (6,38), (1,274403)

Edited on July 3, 2016, 7:52 pm
  Posted by Brian Smith on 2016-07-03 19:51:22

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