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?1.Introductiona + 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.2.MethodSmallest 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]3.Results[1]: MathOverfow: Rational solution to a + b + c = abc = 6.4.Reference

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