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
Smallest c for 1,2,3,4,5,6 distinct solutions
