1.Introduction

It seems that Euler treated the problem to find three positive numbers
such that the sum and difference of any two are squares.([1].Dickson)
Euler gave a solution as {434657, 420968, 150568} and {2843458, 2040642, 1761858}.  
And furthermore, he gave a parametric solution as follows.

A=(8f^6g^2+72f^2g^6)^2+(2f^4g^4-f^8-81g^8)^2
B=2(8f^6g^2+72f^2g^6)*(2f^4g^4-f^8-81g^8)
C=32(30f^4g^4+f^8+81g^8)f^4g^4


I show some numeric solutions and a parametric solution.


2. Method

Simultaneous equation {A + B = p^2, B + C = q^2, C + A = r^2, A - B = u^2, B - C = v^2, A - C = w^2}
has infinitely many solutions.


(i). Method1

A + B = p^2............................................................................................(1)  
B + C = q^2............................................................................................(2)  
C + A = r^2............................................................................................(3)  
A - B = u^2............................................................................................(4)  
B - C = v^2............................................................................................(5)  
A - C = w^2............................................................................................(6)  
   
From (1),(2), and (3), A =(p^2 + r^2 - q^2)/2, B =(p^2 + q^2 - r^2)/2, C =(q^2 + r^2 - p^2)/2,.........(7)


From (4), (5), and (6), r^2 - q^2 = u^2, p^2 - r^2 = v^2, p^2 - q^2 = w^2..............................(8)   


Hence, p^2 = (a^2+b^2)(c^2+d^2) = r^2 + v^2 = q^2 + w^2................................................(9)
We use famous identity as follows.

(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2 = (ac - bd)^2 + (ad + bc)^2.

Take r = (ac+bd), v = (ad-bc), q = (ad+bc), w = (ac-bd),
And furthermore, take a = (ms+nt), b = mt-ns, c = (ms-nt), d  =mt+ns.

Hence, p = (t^2+s^2)(m^2+n^2), q = 2st(m^2+n^2), r = (t^2+s^2)(m^2-n^2),  
v = 2nm(t^2+s^2), w = (s^2-t^2)(m^2+n^2)

u^2 = (t^4-2t^2s^2+s^4)n^4+(-12t^2s^2-2t^4-2s^4)m^2n^2+(t^4-2t^2s^2+s^4)m^4...........................(10)
If we find the integer solutions of (10), we can solve a problem.
  
Accordingly, we must find the integer solutions of (10).


(ii). Method2

Set A= a^2x^2+1, B= x^2+a^2, C= 2ax..................................................................(11)

A + B = (a^2+1)x^2+a^2+1 ............................................................................(12)  
B + C = (x+a)^2......................................................................................(13)  
C + A = (ax+1)^2.....................................................................................(14)  
A - B = (a^2-1)x^2-a^2+1.............................................................................(15)  
B - C = (x-a)^2......................................................................................(16)  
A - C = (ax-1)^2.....................................................................................(17)  

Then A + B and  A - B must be squares.
Here, we consider (A + B)(A - B) be square.

(a^4-1)(x^4-1)=y^2, let x=t-a and y=gt^2+ht+a^4-1, then we obtain x = a(a^8-3+6a^4)/(3a^8-6a^4-1).

From (11), we obtain a parametric solution for {A,B,C} as follows.

A= a^20+21a^16-6a^12-6a^8+21a^4+1
B= 10a^18-24a^14+60a^10-24a^6+10a^2
C= 6a^18+24a^14-92a^10+24a^6+6a^2

A + B = (a^2+1)^2(a^8+4a^6-6a^4+4a^2+1)^2
B + C = 16a^2(a-1)^2(a+1)^2(a^2+1)^2(a^4+1)^2
C + A = (a-1)^2(a+1)^2(a^4-2a^3+4a^2-2a+1)^2(a^4+2a^3+4a^2+2a+1)^2
A - B = (a-1)^2(a+1)^2(a^4-2a^3-2a+1)^2(a^4+2a^3+2a+1)^2
B - C = 4a^2(a^4-2a^2-1)^2(a^4+2a^2-1)^2
A - C = (a^2+1)^2(a^4-2a^3+2a+1)^2(a^4+2a^3-2a+1)^2


3. Results

{A,B,C}<10^8


(1). Brute force.

{p,q,r}<10000
Search for the solution of {r^2 - q^2 = u^2, p^2 - r^2 = v^2, p^2 - q^2 = w^2}.

[    p      q       r      u      v      w ] {      A        B         C    ]

[    925    765    756    117    533    520] [    420968    434657    150568]
[   1105   1073    952    495    561    264] [    488000    733025    418304]
[   1394   1360   1344    208    370    306] [    949986    993250    856350]
[   2165   2067   2040    333    725    644] [   2288168   2399057   1873432]
[   2210   2146   1950    896   1040    528] [   2040642   2843458   1761858]
[   2146   1950    990   1680   1904    896] [    891458   3713858     88642]
[   2665   2431   1540   1881   2175   1092] [   1782032   5320193    589568]
[   3485   3179   2640   1771   2275   1428] [   4504392   7640833   2465208]
[   4225   4199   4180    399    615    468] [   8845712   9004913   8626688]
[   3965   3875   1364   3627   3723    840] [   1283048  14438177    577448]
[   6970   5304   5280    504   4550   4522] [  24163442  24417458   3714958]
[   6970   6888   6720   1512   1850   1066] [  23147378  25433522  22011022]
[   6554   5304   4104   3360   5110   3850] [  15832658  27122258   1010158]
[   8362   8200   7480   3360   3738   1638] [  29316722  40606322  26633678]
[   9061   8925   8736   1827   2405   1564] [  39381896  42719825  36935800]
[   8906   7656   5544   5280   6970   4550] [  25719218  53597618   5016718]
[   9773   9435   8352   4389   5075   2548] [  38124104  57387425  31631800]
[   9997   9605   7800   5605   6253   2772] [  34261992  65678017  26578008]

(2). Method1.

{m,n,s,t}<100
Search for the solution of (10).

[  m   n   s   t ] [    p      q       r      u      v      w ] [      A        B         C    ]

[  39  12  27  14] [    925    756    765    117    520    533] [    434657    420968    150568]
[  33   4   7   4] [   1105    952   1073    495    264    561] [    733025    488000    418304]
[  27   3  21  16] [   1394   1344   1360    208    306    370] [    993250    949986    856350]
[  46   7  17  12] [   2165   2040   2067    333    644    725] [   2399057   2288168   1873432]
[  33   4   5   3] [   2210   1950   2146    896    528   1040] [   2843458   2040642   1761858]
[  64  14  45  11] [   2146    990   1950   1680    896   1904] [   3713858    891458     88642]
[  28   6  22   7] [   2665   1540   2431   1881   1092   2175] [   5320193   1782032    589568]
[  28   6  24  11] [   3485   2640   3179   1771   1428   2275] [   7640833   4504392   2465208]
[  36   2  22  19] [   4225   4180   4199    399    468    615] [   9004913   8845712   8626688]
[  84   9  62  11] [   3965   1364   3875   3627    840   3723] [  14438177   1283048    577448]
[  38  14  24  11] [   6970   5280   5304    504   4522   4550] [  24417458  24163442   3714958]
[  26   2  21  16] [   6970   6720   6888   1512   1066   1850] [  25433522  23147378  22011022]
[  77  25  54  19] [   6554   4104   5304   3360   3850   5110] [  27122258  15832658   1010158]
[  91   9  55  34] [   8362   7480   8200   3360   1638   3738] [  40606322  29316722  26633678]
[  23   2  21  16] [   9061   8736   8925   1827   1564   2405] [  42719825  39381896  36935800]
[  91  25  63  22] [   8906   5544   7656   5280   4550   6970] [  53597618  25719218   5016718]
[  98  13  16   9] [   9773   8352   9435   4389   2548   5075] [  57387425  38124104  31631800]
[  99  14  25  12] [   9997   7800   9605   5605   2772   6253] [  65678017  34261992  26578008]
[  32   9  23  11] [  11050   8602   9430   3864   5760   6936] [  68516498  53586002  20408402]

These solutions contain the solutions of brute force.

(3). Method2.

a<10

    a                  A                   B                   C
    
    2  [             2399057             2288168             1873432  ]
    3  [           274221202           235183698           152118702  ]
    4  [       1189604889857        680815132832        418662940768  ]
    5  [       6160644385586       2375067115250       1439610619150  ]
    6  [    3715388677375057    1013722453597032     611234918180568  ]
    7  [    5030629672332002    1016742223190498     611670809769502  ]
    8  [ 1158832066700333057  180038496396771968  108191845395202432  ]
    9  [  762286074261716002   93774844946586402   56319783376843998  ]







    
    
4. References

[1].L.E. Dickson:History of theory of numbers, vol 2.
    
    
    
    


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