1.Introduction

We show numeric solutions for {  x^2+A=u^2, x^2-A=v^2 }.





2.Method

x^2+A=u^2.............................................................(1)

x^2-A=v^2.............................................................(2)

From equation (1), we obtain (x,u)=(1/2(At^2-1)/t, 1/2(At^2+1)/t).....(3)
Substitute equation (3) to equation (2) and V=2tv, we obtain
V^2 = A^2t^4-6At^2+1..................................................(4)
We searched the rational solutions of equation (4).
A<100
x>0
height of t<100000


3.Results

[A, 0, 0, 0]: No solution.

[A, x, u, v]

[1, 0, 0, 0]
[2, 0, 0, 0]
[3, 0, 0, 0]
[4, 0, 0, 0]
[5, 41/12, 49/12, 31/12]
[5, 3344161/1494696, 4728001/1494696, 113279/1494696]
[6, 5/2, 7/2, 1/2]
[6, 1201/140, 1249/140, 1151/140]
[6, 7776485/2639802, 10113607/2639802, 4319999/2639802]
[7, 337/120, 463/120, 113/120]
[7, 23058557761/4231560720, 25632788161/4231560720, 20158232639/4231560720]
[8, 0, 0, 0]
[9, 0, 0, 0]
[10, 0, 0, 0]
[11, 0, 0, 0]
[12, 0, 0, 0]
[13, 106921/19380, 127729/19380, 80929/19380]
[14, 65/12, 79/12, 47/12]
[14, 21914881/5792280, 30821569/5792280, 3248831/5792280]
[15, 17/4, 23/4, 7/4]
[15, 141121/21896, 164641/21896, 112799/21896]
[15, 440084695697/45021381108, 473369211863/45021381108, 404067652793/45021381108]
[16, 0, 0, 0]
[17, 0, 0, 0]
[18, 0, 0, 0]
[19, 0, 0, 0]
[20, 41/6, 49/6, 31/6]
[20, 3344161/747348, 4728001/747348, 113279/747348]
[21, 25/4, 31/4, 17/4]
[21, 503521/105400, 697729/105400, 142271/105400]
[22, 4901/210, 4999/210, 4801/210]
[23, 905141617/144613560, 1140299183/144613560, 581618833/144613560]
[24, 5, 7, 1]
[24, 1201/70, 1249/70, 1151/70]
[24, 7776485/1319901, 10113607/1319901, 4319999/1319901]
[25, 0, 0, 0]
[26, 0, 0, 0]
[27, 0, 0, 0]
[28, 337/60, 463/60, 113/60]
[28, 23058557761/2115780360, 25632788161/2115780360, 20158232639/2115780360]
[29, 48029801/180180, 48039601/180180, 48019999/180180]
[30, 13/2, 17/2, 7/2]
[30, 42961/6188, 54721/6188, 26399/6188]
[30, 34597174573/455676474, 34687082897/455676474, 34507031993/455676474]
[31, 2566561/206640, 2812639/206640, 2294239/206640]
[32, 0, 0, 0]
[33, 0, 0, 0]
[34, 145/12, 161/12, 127/12]
[34, 353/60, 497/60, 47/60]
[34, 3425/504, 4513/504, 1759/504]
[34, 939905/114576, 1153153/114576, 661121/114576]
[34, 818472833/60857940, 892089233/60857940, 737544817/60857940]
[34, 17371409905/881464188, 18115826561/881464188, 16593631073/881464188]
[34, 466021441/71155560, 623956609/71155560, 212203009/71155560]
[34, 30509162881/989487240, 31049926081/989487240, 29958640319/989487240]
[35, 0, 0, 0]
[36, 0, 0, 0]
[37, 605170417321/9475102140, 607908713329/9475102140, 602419674529/9475102140]
[38, 1646021/237150, 2201479/237150, 756479/237150]
[39, 313/20, 337/20, 287/20]
[39, 9841284961/1210921880, 12411197761/1210921880, 6297932161/1210921880]
[40, 0, 0, 0]
[41, 881/120, 1169/120, 431/120]
[41, 1054721/81840, 1177729/81840, 915329/81840]
[41, 354481/24360, 387281/24360, 318319/24360]
[41, 106088910401/16481869920, 149641539649/16481869920, 10822357951/16481869920]
[42, 0, 0, 0]
[43, 0, 0, 0]
[44, 0, 0, 0]
[45, 41/4, 49/4, 31/4]
[45, 3344161/498232, 4728001/498232, 113279/498232]
[46, 7585/924, 9839/924, 4273/924]
[47, 217287944875297/24222075378480, 273476454129503/24222075378480, 140138256778847/24222075378480]
[48, 0, 0, 0]
[49, 0, 0, 0]
[50, 0, 0, 0]
[51, 0, 0, 0]
[52, 106921/9690, 127729/9690, 80929/9690]
[53, 4850493897329785961/595711308569957580, 6506573990620136689/595711308569957580, 2172343665411286111/595711308569957580]
[54, 15/2, 21/2, 3/2]
[54, 3603/140, 3747/140, 3453/140]
[54, 7776485/879934, 10113607/879934, 4319999/879934]
[55, 17561/2340, 24689/2340, 2689/2340]
[56, 65/6, 79/6, 47/6]
[56, 21914881/2896140, 30821569/2896140, 3248831/2896140]
[57, 0, 0, 0]
[58, 0, 0, 0]
[59, 0, 0, 0]
[60, 17/2, 23/2, 7/2]
[60, 141121/10948, 164641/10948, 112799/10948]
[60, 440084695697/22510690554, 473369211863/22510690554, 404067652793/22510690554]
[61, 250510625883241/18295510698660, 288398755364209/18295510698660, 205760310228191/18295510698660]
[62, 2056525601/242137560, 2804352799/242137560, 770844001/242137560]
[63, 337/40, 463/40, 113/40]
[63, 23058557761/1410520240, 25632788161/1410520240, 20158232639/1410520240]
[64, 0, 0, 0]
[65, 97/12, 137/12, 7/12]
[65, 4481/504, 6049/504, 1889/504]
[65, 14657/660, 15593/660, 13657/660]
[65, 23433857/759684, 24221033/759684, 22619303/759684]
[65, 176138881/2232552, 177056161/2232552, 175216799/2232552]
[65, 510228161/53247264, 666802369/53247264, 275753791/53247264]
[66, 0, 0, 0]
[67, 0, 0, 0]
[68, 0, 0, 0]
[69, 65425/3952, 73199/3952, 56593/3952]
[70, 53/6, 73/6, 17/6]
[70, 14240881/789276, 15697441/789276, 12617279/789276]
[71, 12974641/563640, 13816559/563640, 12074159/563640]
[72, 0, 0, 0]
[73, 0, 0, 0]
[74, 0, 0, 0]
[75, 0, 0, 0]
[76, 0, 0, 0]
[77, 15820337/127200, 15859663/127200, 15780913/127200]
[78, 677/30, 727/30, 623/30]
[79, 56434050774922081/991366231674000, 57117806683077919/991366231674000, 55741908232922081/991366231674000]
[80, 41/3, 49/3, 31/3]
[80, 3344161/373674, 4728001/373674, 113279/373674]
[81, 0, 0, 0]
[82, 0, 0, 0]
[83, 0, 0, 0]
[84, 25/2, 31/2, 17/2]
[84, 503521/52700, 697729/52700, 142271/52700]
[84, 9473784254425/360836330118, 10034422160671/360836330118, 8877812123503/360836330118]
[85, 8521/924, 12049/924, 191/924]
[86, 116645/9282, 144967/9282, 78719/9282]
[87, 322446497/6706700, 328458503/6706700, 316320247/6706700]
[88, 4901/105, 4999/105, 4801/105]
[89, 0, 0, 0]
[90, 0, 0, 0]
[91, 0, 0, 0]
[92, 905141617/72306780, 1140299183/72306780, 581618833/72306780]
[93, 2090761/98420, 2296111/98420, 1862911/98420]
[94, 199428385/7948248, 213799199/7948248, 183938207/7948248]
[95, 2093801/98124, 2301889/98124, 1862609/98124]
[96, 10, 14, 2]
[96, 1201/35, 1249/35, 1151/35]
[96, 15552970/1319901, 20227214/1319901, 8639998/1319901]
[97, 0, 0, 0]
[98, 0, 0, 0]
[99, 0, 0, 0]