Rational Triangles Whose Area Equals a Side. Data by James R. Buddenhagen, Feb. 6, 2012. ----------------------------------------------------------------------------------------- [S,T] rational parameters, Triangle = [S/T + T/S, 1/(S*T) + S*T, S/T - T/S + 1/(S*T) - S*T] has third side equal to its area. We need the third side, namely, S/T - T/S + 1/(S*T) - S*T to be an integer. ----------------------------------------------------------------------------------------- Note: As of Feb. 6, 2012, triangles for N=127 and for N=139 have not been found, but are conjectured to exist, based on standard elliptic curve conjectures as determined within Ian Connell's program APECS. ----------------------------------------------------------------------------------------- N = 1. No triangle exists, per APECS N = 2. No triangle exists, per APECS N = 3. [S,T] = [1, 1/2], Triangle = [5/2, 5/2, 3] N = 4. No triangle exists, per APECS N = 5. [S,T] = [3, 1/2], Triangle = [37/6, 13/6, 5] N = 6. No triangle exists, per APECS N = 7. [S,T] = [3, 2/5], Triangle = [229/30, 61/30, 7] N = 8. No triangle exists, per APECS N = 9. No triangle exists, per APECS N = 10. [S,T] = [4/3, 1/5], Triangle = [409/60, 241/60, 10] N = 11. No triangle exists, per APECS N = 12. [S,T] = [2, 1/5], Triangle = [101/10, 29/10, 12] N = 13. No triangle exists, per APECS N = 14. No triangle exists, per APECS N = 15. [S,T] = [7, 2/5], Triangle = [1229/70, 221/70, 15] N = 16. [S,T] = [3, 1/5], Triangle = [226/15, 34/15, 16] N = 17. No triangle exists, per APECS N = 18. [S,T] = [8, 5/13], Triangle = [10841/520, 1769/520, 18] N = 19. [S,T] = [5/3, 2/17], Triangle = [7261/510, 2701/510, 19] N = 20. No triangle exists, per APECS N = 21. No triangle exists, per APECS N = 22. No triangle exists, per APECS N = 23. [S,T] = [55/3, 41/74], Triangle = [16580029/500610, 5134309/500610, 23] N = 24. No triangle exists, per APECS N = 25. [S,T] = [11/3, 2/13], Triangle = [20485/858, 2005/858, 25] N = 26. [S,T] = [33/4, 5/17], Triangle = [315121/11220, 31849/11220, 26] N = 27. [S,T] = [55/7, 29/106], Triangle = [34030109/1183490, 3094589/1183490, 27] N = 28. [S,T] = [3/2, 1/13], Triangle = [1525/78, 685/78, 28] N = 29. [S,T] = [273/115, 17/178], Triangle = [2365198861/95001270, 440559781/95001270, 29] N = 30. [S,T] = [7/4, 1/13], Triangle = [8297/364, 2753/364, 30] N = 31. No triangle exists, per APECS N = 32. No triangle exists, per APECS N = 33. [S,T] = [3, 1/10], Triangle = [901/30, 109/30, 33] N = 34. [S,T] = [689/168, 61/485], Triangle = [111771268729/3424522920, 8405427241/3424522920, 34] N = 35. No triangle exists, per APECS N = 36. [S,T] = [35/4, 17/73], Triangle = [6532649/173740, 439289/173740, 36] N = 37. No triangle exists, per APECS N = 38. [S,T] = [1067/696, 305/5321], Triangle = [32279148533449/1205221185960, 13821198008281/1205221185960, 38] N = 39. [S,T] = [7/5, 2/37], Triangle = [67181/2590, 34421/2590, 39] N = 40. [S,T] = [18, 5/13], Triangle = [54781/1170, 8269/1170, 40] N = 41. [S,T] = [640/21, 73/137], Triangle = [7690132489/134413440, 2191035529/134413440, 41] N = 42. [S,T] = [55/16, 17/193], Triangle = [112752209/2887280, 10409969/2887280, 42] N = 43. [S,T] = [322287/1993849, 767890/5312353], Triangle = [5275428969376154806670821/2621325729559393234087710, 112252340347147533005814709/2621325729559393234087710, 43] N = 44. [S,T] = [35/6, 13/97], Triangle = [11532109/264810, 545749/264810, 44] N = 45. No triangle exists, per APECS N = 46. [S,T] = [464/75, 173/1277], Triangle = [351257781409/7688050800, 15616444609/7688050800, 46] N = 47. [S,T] = [21, 13/34], Triangle = [509965/9282, 75685/9282, 47] N = 48. No triangle exists, per APECS N = 49. No triangle exists, per APECS N = 50. No triangle exists, per APECS N = 51. [S,T] = [16/13, 1/25], Triangle = [160169/5200, 105881/5200, 51] N = 52. [S,T] = [4/3, 1/25], Triangle = [10009/300, 5641/300, 52] N = 53. No triangle exists, per APECS N = 54. No triangle exists, per APECS N = 55. [S,T] = [9177/7991, 1042/28421], Triangle = [68096153082044173/2171745528281574, 51671396529312877/2171745528281574, 55] N = 56. No triangle exists, per APECS N = 57. [S,T] = [8339/4697, 1937/47290], Triangle = [155595730922168021/3587843315646590, 49598704268362349/3587843315646590, 57] N = 58. [S,T] = [12, 1/5], Triangle = [3601/60, 169/60, 58] N = 59. [S,T] = [1725/13717, 5069/37706], Triangle = [9065198073672229/4522525168618050, 267585954296293429/4522525168618050, 59] N = 60. No triangle exists, per APECS N = 61. No triangle exists, per APECS N = 62. [S,T] = [361073/16696344, 12199909/22860685], Triangle = [41559330428614145828995978441/1681365876653201851843203480, 145706554398443729268886125049/1681365876653201851843203480, 62] N = 63. [S,T] = [377/11, 250/569], Triangle = [46023389669/589910750, 8922237581/589910750, 63] N = 64. [S,T] = [567/391, 89/2665], Triangle = [2284498183426/52583202945, 1088341774594/52583202945, 64] N = 65. [S,T] = [29/3, 5/34], Triangle = [972421/14790, 31429/14790, 65] N = 66. No triangle exists, per APECS N = 67. [S,T] = [9/7, 2/65], Triangle = [342421/8190, 207349/8190, 67] N = 68. [S,T] = [12, 5/29], Triangle = [121129/1740, 4441/1740, 68] N = 69. [S,T] = [320/2821, 1013/7957], Triangle = [14649633112529/7276321299520, 503959285915409/7276321299520, 69] N = 70. [S,T] = [5757/3596, 541/17033], Triangle = [9619348475988457/190767471760716, 3761344671170593/190767471760716, 70] N = 71. [S,T] = [347996593/505868235, 106589306/3536980529], Triangle = [1517916763740293774941675148773171909/66367996162188396246751100052822270, 3202777466716096223779488555590132989/66367996162188396246751100052822270, 71] N = 72. [S,T] = [4, 1/17], Triangle = [4625/68, 305/68, 72] N = 73. No triangle exists, per APECS N = 74. [S,T] = [5019/124028, 11245/37033], Triangle = [1979720899013284729/259230018712157220, 21100018634263606801/259230018712157220, 74] N = 75. [S,T] = [11/7, 1/34], Triangle = [139925/2618, 56765/2618, 75] N = 76. [S,T] = [11070/326599, 191369/558029], Triangle = [3944525541118637149861/386092022466570531930, 33220174877531890512541/386092022466570531930, 76] N = 77. [S,T] = [5/3, 1/34], Triangle = [28909/510, 10429/510, 77] N = 78. No triangle exists, per APECS N = 79. No triangle exists, per APECS N = 80. [S,T] = [7/3, 1/29], Triangle = [41218/609, 7618/609, 80] N = 81. No triangle exists, per APECS N = 82. No triangle exists, per APECS N = 83. [S,T] = [479273635575/867812145607, 167203816793/5877863057618], Triangle = [7957133883660679377014027726558287383216156401701/408766140172074379875594526759246516591807982850, 26025408322139692495797954909903424034387955634501/408766140172074379875594526759246516591807982850, 83] N = 84. [S,T] = [5/2, 1/29], Triangle = [21029/290, 3389/290, 84] N = 85. [S,T] = [21/16, 1/41], Triangle = [741577/13776, 430777/13776, 85] N = 86. [S,T] = [904/45465, 30869/66989], Triangle = [1973364681415536361/84990753863138760, 9276792399696255001/84990753863138760, 86] N = 87. [S,T] = [13, 5/34], Triangle = [195389/2210, 5381/2210, 87] N = 88. [S,T] = [1199/348, 257/6065], Triangle = [52889037442321/650372778660, 4549668892849/650372778660, 88] N = 89. [S,T] = [5675/9792, 193/7457], Triangle = [1794424882026361/79975823889600, 5332965810200761/79975823889600, 89] N = 90. [S,T] = [18067/52756, 4745/131069], Triangle = [5670199596158974529/592780770439918060, 47820095728400002121/592780770439918060, 90] N = 91. [S,T] = [1037/249, 65/1346], Triangle = [1948525177429/22591055370, 116871637741/22591055370, 91] N = 92. No triangle exists, per APECS N = 93. [S,T] = [377/23329, 16490/33013], Triangle = [148145305048645901/4787878498834210, 593185500000609629/4787878498834210, 93] N = 94. No triangle exists, per APECS N = 95. [S,T] = [183/17711, 6253/10034], Triangle = [12268243805448973/203355862996026, 31582926208007677/203355862996026, 95] N = 96. [S,T] = [17, 5/29], Triangle = [243074/2465, 8066/2465, 96] N = 97. [S,T] = [48/7, 13/181], Triangle = [75489625/790608, 1994665/790608, 97] N = 98. No triangle exists, per APECS N = 99. No triangle exists, per APECS N = 100. No triangle exists, per APECS N = 101. No triangle exists, per APECS N = 102. No triangle exists, per APECS N = 103. [S,T] = [11/9, 2/101], Triangle = [1234645/19998, 826765/19998, 103] N = 104. [S,T] = [2394/2987, 149/7565], Triangle = [328192275646069/8060362418430, 510736056359461/8060362418430, 104] N = 105. [S,T] = [8, 1/13], Triangle = [10817/104, 233/104, 105] N = 106. [S,T] = [357/4, 37/65], Triangle = [538493929/3434340, 174545281/3434340, 106] N = 107. [S,T] = [378873/14039, 185050/776773], Triangle = [86618248460861561917741/764562585646529351550, 5034396568083846948109/764562585646529351550, 107] N = 108. No triangle exists, per APECS N = 109. [S,T] = [60225/81613,1346/70121], Triangle = [17846110667467506229/463905276061123050, 32756841321958762429/463905276061123050, 109] N = 110. No triangle exists, per APECS N = 111. No triangle exists, per APECS N = 112. No triangle exists, per APECS N = 113. No triangle exists, per APECS N = 114. No triangle exists, per APECS N = 115. [S,T] = [194163/158441, 14930/841297], Triangle = [26688409638547285103821/386405408337043787430, 17776210433304486740629/386405408337043787430, 115] N = 116. No triangle exists, per APECS N = 117. No triangle exists, per APECS N = 118. No triangle exists, per APECS N = 119. [S,T] = [313872/96241, 13781/460045], Triangle = [20851745623447685500441/191511011161128257040, 1979002215358901895049/191511011161128257040, 119] N = 120. No triangle exists, per APECS N = 121. No triangle exists, per APECS N = 122. [S,T] = [28/3, 1/13], Triangle = [132505/1092, 2305/1092, 122] N = 123. [S,T] = [55, 34/89], Triangle = [23962181/166430, 3504821/166430, 123] N = 124. [S,T] = [6/5, 1/61], Triangle = [133981/1830, 93061/1830, 124] N = 125. [S,T] = [693/47009, 13922/31753], Triangle = [428801185237972045/14401264939950042, 2228177043457126645/14401264939950042, 125] N = 126. [S,T] = [221/80, 37/1493], Triangle = [108877743809/976660880, 14332776929/976660880, 126] N = 127. Triangle exists, per APECS (standard elliptic curve conjectures) but none found. N = 128. No triangle exists, per APECS N = 129. No triangle exists, per APECS N = 130. No triangle exists, per APECS N = 131. [S,T] = [23440/81933, 14473/501793], Triangle=[139751461109549909881/13947604534449347280, 1690427230757551939561/13947604534449347280, 131] N = 132. No triangle exists, per APECS N = 133. [S,T] = [15293/69285, 13402/375637], Triangle = [33862839512858427781/5334207268748522370, 677395180120558556821/5334207268748522370, 133] N = 134. No triangle exists, per APECS N = 135. [S,T] = [5, 1/26], Triangle = [16901/130, 701/130, 135] N = 136. [S,T] = [45/4, 13/157], Triangle = [49916929/367380, 736609/367380, 136] N = 137. [S,T] = [33/16, 5/269], Triangle = [78807529/710160, 18551641/710160, 137] N = 138. [S,T] = [377/56, 85/1709], Triangle = [415136127449/3066834680, 10186137641/3066834680, 138] N = 139. Triangle exists, per APECS (standard elliptic curve conjectures) but none found. N = 140. No triangle exists, per APECS N = 141. [S,T] = [793/175, 149/4426], Triangle = [12319502298749/91518504350, 613888779149/91518504350, 141] N = 142. [S,T] = [984/30535, 13973/66797], Triangle = [186363698408607529/28043979412097640, 4160346463516818649/28043979412097640, 142] N = 143. No triangle exists, per APECS N = 144. [S,T] = [1253/391, 145/5941], Triangle = [55417437152354/422041718735, 5429017869986/422041718735, 144] N = 145. [S,T] = [27/5, 1/26], Triangle = [492829/3510, 17629/3510, 145]