1.Introduction

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

Two square theorem.
"Any positive number n is expressible as a sum of two squares if and only if the prime factorization of n,
 every prime of the form (4k + 3) have even exponent.h
 
From second equation A=x^2+v^2, we can exclude below case by two square theorem.

1) A=2,3 mod 4.

2) A has prime factors p=3 mod 4 with odd exponent.


No solution was found for A=17,73,89,97.
There are infinietly many rational solutions for A=5,13,20,29,37,41,52,53,61,65,80,85.

There is another parametric solution for { A+x^2=u^2, A-x^2=v^2 }.
It's different from last time.

(A,x,u,v)=( (16+k^2)(k^2-16k+80), 8k-32, k^2-8k+48, k^2-8k-16)

2.Method

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

A-x^2=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<1000000


3.Results

[A, ?, ?, ?]: No solution was found(Rank is unknown).
[A, 0, 0, 0]: Rank is 0.

[A, x, u, v]

[1, 0, 0, 0]
[2, -, -, -]
[3, -, -, -]
[4, 0, 0, 0]
[5, 2, 3, 1]
[5, 1562/2257, 5283/2257, 4799/2257]
[5, 7118599318/8914433905, 21166249443/8914433905, 18618840001/8914433905]
[6, -, -, -]
[7, -, -, -]
[9, -, -, -]
[8, -, -, -]
[9, 0, 0, 0]
[10, -, -, -]
[11, -, -, -]
[12, -, -, -]
[13, 6/5, 19/5, 17/5]
[13, 195881413554/59864708885, 291476777779/59864708885, 90662316497/59864708885]
[14, -, -, -]
[15, -, -, -]
[16, 0, 0, 0]
[17, ?, ?, ?]
[18, -, -, -]
[19, -, -, -]
[20, 4, 6, 2]
[20, 3124/2257, 10566/2257, 9598/2257]
[20, 14237198636/8914433905, 42332498886/8914433905, 37237680002/8914433905]
[21, -, -, -]
[22, -, -, -]
[23, -, -, -]
[24, -, -, -]
[25, 0, 0, 0]
[26, -, -, -]
[27, -, -, -]
[28, -, -, -]
[29, 70/13, 99/13, 1/13]
[30, -, -, -]
[31, -, -, -]
[32, -, -, -]
[33, -, -, -]
[34, -, -, -]
[35, -, -, -]
[36, 0, 0, 0]
[37, 42/145, 883/145, 881/145]
[38, -, -, -]
[39, -, -, -]
[40, -, -, -]
[41, 8/5, 33/5, 31/5]
[41, 368/85, 657/85, 401/85]
[41, 1509472/239125, 2150097/239125, 256721/239125]
[41, 8722551592/1511573765, 13029273633/1511573765, 4194777569/1511573765]
[41, 45913055872/37404678805, 243867830097/37404678805, 235064887121/37404678805]
[42, -, -, -]
[43, -, -, -]
[44, -, -, -]
[45, 6, 9, 3]
[46, -, -, -]
[47, -, -, -]
[48, -, -, -]
[49, 0, 0, 0]
[50, -, -, -]
[51, -, -, -]
[52, 12/5, 38/5, 34/5]
[52, 391762827108/59864708885, 582953555558/59864708885, 181324632994/59864708885]
[53, 34034/5945, 55059/5945, 26737/5945]
[54, -, -, -]
[55, -, -, -]
[56, -, -, -]
[57, -, -, -]
[58, -, -, -]
[59, -, -, -]
[60, -, -, -]
[61, 3198/445, 4723/445, 1361/445]
[62, -, -, -]
[63, -, -, -]
[64, 0, 0, 0]
[65, 4, 9, 7]
[65, 4088/689, 6897/689, 3761/689]
[65, 112/17, 177/17, 79/17]
[65, 14450276/4525537, 39243369/4525537, 33502553/4525537]
[65, 187986956/24143617, 270607689/24143617, 50500793/24143617]
[65, 14257432/13046897, 106149297/13046897, 104216719/13046897]
[65, 32894864276/6873141025, 64441263849/6873141025, 44592962393/6873141025]
[65, 540729884764/67549356961, 767449232169/67549356961, 64812864487/67549356961]
[65, 1552837424768/810420276305, 6715807519857/810420276305, 6346610316401/810420276305]
[66, -, -, -]
[67, -, -, -]
[69, -, -, -]
[70, -, -, -]
[71, -, -, -]
[72, -, -, -]
[73, ?, ?, ?]
[74, -, -, -]
[75, -, -, -]
[76, -, -, -]
[77, -, -, -]
[78, -, -, -]
[79, -, -, -]
[80, 8, 12, 4]
[80, 6248/2257, 21132/2257, 19196/2257]
[80, 28474397272/8914433905, 84664997772/8914433905, 74475360004/8914433905]
[81, 0, 0, 0]
[82, -, -, -]
[83, -, -, -]
[84, -, -, -]
[85, 6, 11, 7]
[85, 592443954/103343377, 1121952491/103343377, 746188793/103343377]
[86, -, -, -]
[87, -, -, -]
[88, -, -, -]
[89, ?, ?, ?]
[90, -, -, -]
[91, -, -, -]
[92, -, -, -]
[93, -, -, -]
[94, -, -, -]
[95, -, -, -]
[96, -, -, -]
[97, ?, ?, ?]
[98, -, -, -]
[99, -, -, -]






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