1.Introduction

It's showed that a + b + c = abc = 6 has infinitely many rational solutions on MathOverfow website[1].

Hence we searched the positive rational solutions of a + b + c = abc = N. 
We found the positive rational solutions for several N, but no positive rational solution was found for N=16,24,28,30,32,33,
40,43,44,48,52,55,56,58,60,62,64,70,72,74,76,82,84,88,91,96.
I don't know whether there are positive rational solutions for remaining N.
Most of remaining N are even number except for N=33, 55, 43, and 91.
Especially, are there positive rational solutions for N=33, 55, 43, 91?

2.Method
      
 
a + b + c = N...................................(1)
abc = N.........................................(2)
Substitute c = N-a-b to equation (2), we oobtain
-ab^2+(aN-a^2)b-N=0.............................(3)
In order to have rational solutions, we must have rational solutions of equation (4).
v^2=a^4-2Na^3+N^2a^2-4aN........................(4)
Hence the problem was reduced to elliptic curve problem.
We can search the rational solution by usual method.


 
3.Results

Smallest solutions are shown, since there are infinitely many rational solutions.(1<=Rank<=3)

[-]: No solution.

N  [a,b,c]
[1][-]
[2][-]
[3][-]
[4][-]
[5][-]
[6][3, 2, 1]
[7][4/3, 9/2, 7/6]
[8][-]
[9][1/2, 9/2, 4]
[10][-]
[11][-]
[12][-]
[13][36/77, 637/66, 121/42]
[14][1/3, 9, 14/3]
[15][1/2, 12, 5/2]
[16][18, -2/3, -4/3]
[17][-]
[18][-]
[19][121/234, 3211/198, 324/143]
[20][-]
[21][-]
[22][1/3, 18, 11/3]
[23][-]
[24][27, -1/3, -8/3]
[25][529/3162, 24025/2346, 10404/713]
[26][-]
[27][4/7, 49/2, 27/14]
[28][81/2, -1/18, -112/9]
[29][2601/658, 1421/4794, 8836/357]
[30][32, -3/4, -5/4]
[31][12321/90818, 249001/20202, 1026844/55389]
[32][648/11, -2/99, -242/9]
[33][704/13, -3/104, -169/8]
[34][-]
[35][-]
[36][-]
[37][81/50, 625/18, 148/225]
[38][9/85, 289/15, 950/51]
[39][-]
[40][128/3, -5/12, -9/4]
[41][-]
[42][-]
[43][-203131162995712/16466222231841, 157821958923841/2848815393312, -1717989539841/27304722337312]
[44][99/2, -1/6, -16/3]
[45][121/266, 8820/209, 361/154]
[46][2/21, 207/7, 49/3]
[47][-]
[48][50, -4/5, -6/5]
[49][1444/2193, 2601/1634, 90601/1938]
[50][-]
[51][-]
[52][46535632/783591, -3073009/422862, -199809/1658338]
[53][5467286564974404/367112110674973, 186222240590929/1989156057952698, 38356744999728653/1009024456632354]
[54][-]
[55][-64/759, -2645/264, 11979/184]
[56][175/3, -8/15, -9/5]
[57][-]
[58][-3774793149/78659672, 2625515648/24768939, -4713241/413370888]
[59][64025346476/44649770637, 10426656321/14404449514, 191202429289/3363740562]
[60][64, -1/4, -15/4]
[61][61999876/1236501, 1067089/9425178, 87401349/8133842]
[62][775/12, -9/20, -32/15]
[63][1/10, 252/5, 25/2]
[64][19600/279, -1922/315, -162/1085]
[65][-]
[66][-]
[67][-]
[68][-]
[69][3/22, 121/2, 92/11]
[70][72, -5/6, -7/6]
[71][72361/149058, 194481/90922, 8111324/118629]
[72][-8192/34681, 192721/2528, -56169/14048]
[73][1/18, 81/2, 292/9]
[74][31329/406, -14504/5133, -841/2478]
[75][5041/986, 867/4118, 84100/1207]
[76][-50357296/5261865, 51251281/598290, -540225/5827426]
[77][51648597/101878586, 30713764/15055677, 337934689/4538898]
[78][2/5, 75, 13/5]
[79][4356/6059, 6889/4818, 420991/5478]
[80][-]
[81][-]
[82][-24649/78390, 4489992/52595, -112225/36738]
[83][648160681/8006058, 32103556/35973567, 165715227/144250694]
[84][175/2, -3/10, -16/5]
[85][-]
[86][2/9, 81, 43/9]
[87][4747692/113521, 107909/2341138, 3463321/76738]
[88][63081847921/688554135, -15382246275/4624627493, -2712308552/9392165595]
[89][-]
[90][-]
[91][-732736/2292003, 146081299/1548504, -3272481/1084552]
[92][-]
[93][-]
[94][49/93, 1922/21, 423/217]
[95][27556/2163, 4655/51294, 95481/1162]
[96][98, -6/7, -8/7]
[97][6889/9282, 10404/7553, 803257/8466]
[98][-]
[99][49/26, 676/7, 99/182]



4.Reference

[1]: MathOverfow: Rational solution to  a + b + c = abc = 6.