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**