1.Introduction

It seems that Euler teated the problrem to find four positive numbers
such that the sum of any two is a square.([1].Dickson)
Euler gave a solution as {722, 432242, 2814962, 3246482}.  

We show more numeric solutions and a parametric solution.


2. Theorem

Simultaneous equation {A + B = p^2, A + C = q^2, B + D = s^2, C + D = t^2, B + C = u^2, A + D = v^2}
has infinitely many solutions.


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

Similarly, D =(s^2 + t^2 - u^2)/2..........................................(8)

From (1), (3), and B = p^2 - A = s^2 - D, then p^2 - s^2 = A - D...........(9)   

Similarly, C = q^2 - A = t^2 - D, then q^2 - t^2 = A - D..................(10)   

From (9) and (10), p^2 + t^2 = q^2 + s^2..................................(11)   

From (1)+(2)+(3)+(4)+(5)+(6), p^2 + q^2 + s^2 + t^2 = 2(u^2 + v^2)........(12)   

Hence, p^2 + t^2 = q^2 + s^2 = u^2 + v^2..................................(13)
                      
Accordingly, we must find the integer solutions of (13).

We are very happy if (13) has parametric solutions.
So let use famous identity, (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2 = (ac - bd)^2 + (ad + bc)^2.
Consider (a^2 + b^2)(c^2 + d^2)(e^2 + f^2), then we obtain a following identity.

(ace - bde - bcf - adf)^2 + (bce + ade + acf - bdf)^2 = (ace - bde + bcf + adf)^2 + (bce + ade - acf + bdf)^2
                                                      = (ace + bde - bcf + adf)^2 + (bce - ade + acf + bdf)^2
                                                      = (ace + bde + bcf - adf)^2 + (bce - ade - acf - bdf)^2

Set {p,q,s,t,u,v} as follows by above identity.
p = ace -bde -bcf -adf;
q = ace -bde +bcf +adf;
s = bce +ade -acf +bdf;
t = bce +ade +acf -bdf;
u = ace +bde -bcf +adf;
v = bce -ade +acf +bdf;

From (7), (8), B = p^2 - A, and C = q^2 - A

A = 2a^2c^2e^2+2b^2c^2f^2+2a^2d^2f^2-12ace^2bd+2b^2d^2e^2+4ac^2ebf-4a^2cedf+4b^2decf-4bd^2eaf+12bcf^2ad

B = 2a^2c^2e^2+4ace^2bd-12ac^2ebf-4a^2cedf+2b^2d^2e^2+4b^2decf+12bd^2eaf+2b^2c^2f^2-4bcf^2ad+2a^2d^2f^2

C = 2a^2c^2e^2+4ace^2bd+4ac^2ebf+12a^2cedf+2b^2d^2e^2-12b^2decf-4bd^2eaf+2b^2c^2f^2-4bcf^2ad+2a^2d^2f^2

D =-2a^2c^2e^2-2b^2c^2f^2-2a^2d^2f^2+4ace^2bd-2b^2d^2e^2+4ac^2ebf-4a^2cedf+4b^2decf-4bd^2eaf-4bcf^2ad+4b^2c^2e^2+4a^2d^2e^2+4a^2c^2f^2+4b^2d^2f^2

Accordingly, simultaneous equation {A + B = p^2, A + C = q^2, B + D = s^2, C + D = t^2, B + C = u^2, A + D = v^2}
has infinitely many solutions.


3. Results

{A,B,C,D}<10000

(1). Using parametric solution.
{a,b,c,d,e,f}<10

[  a  b  c  d  e  f] [  A      B     C      D ]  

[  1  2  3  7  7  3] [  3362   482   359     2]
[  1  2  3  2  8  5] [  3473  1288  1016     8]
[  1  3  3  5  8  3] [  4194  1282   567   162]
[  1  2  4  3  7  4] [  4743  2482   882    18]
[  4  3  4  7  8  9] [  5024  2720  1376   305]
[  1  2  7  3  9  5] [  5378  1346   863    98]
[  1  2  5  3  3  2] [  5378  2018  1346    98]
[  2  3  1  7  1  9] [  5522  4082  3314   407]
[  3  4  6  7  9  8] [  5634  3775  2466   450]
[  1  2  1  4  8  5] [  6722   674   167     2]
[  1  4  1  2  5  2] [  6722  2114  1022     2]
[  1  4  2  3  7  2] [  6952  2457  1512   792]
[  1  5  5  4  7  1] [  7394  4487  1442     2]
[  1  2  5  2  8  3] [  8073  1728   576   208]
[  1  4  3  4  6  1] [  8114  3767  3122   722]
[  1  3  5  7  8  3] [  8456  4088   953     8]
[  1  5  1  7  5  4] [  8498  3383  1106   338]
[  2  1  3  4  7  8] [  8514  3586  1890   135]
[  1  5  5  6  7  1] [  9097  5544  3672  1512]

Almost solutions were found by using parametric solution except [  6048  1696  1008   513],
[  7697  2912  1712   224], [  8258  4738  1346    23], [  8786  4439  2450    50], [  9800  5576  1649   200].

(2). Brute force.

[  A      B     C      D ]

[  3362   482   359     2]
[  3473  1288  1016     8]
[  4194  1282   567   162]
[  4743  2482   882    18]
[  5024  2720  1376   305]
[  5378  1346   863    98]
[  5378  2018  1346    98]
[  5522  4082  3314   407]
[  5634  3775  2466   450]
[  6048  1696  1008   513]
[  6722   674   167     2]
[  6722  2114  1022     2]
[  6952  2457  1512   792]
[  7394  4487  1442     2]
[  7697  2912  1712   224]
[  8073  1728   576   208]
[  8114  3767  3122   722]
[  8258  4738  1346    23]
[  8456  4088   953     8]
[  8498  3383  1106   338]
[  8514  3586  1890   135]
[  8786  4439  2450    50]
[  9097  5544  3672  1512]
[  9800  5576  1649   200]



    
    
4. Reference

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


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