Generate many examples of Ramanujam's Diophantine Equation

Generate many examples of Ramanujam's Diophantine Equation.

This programme is about finding many examples of Ramanujam's Diophantine Equation. For mathematical formulas used in making this programme, I have referred to the article MATHEMATICAL MINIATURE 9.pdf by John Butcher, [e-mail].

To circumvent the problem of editing the Greek symbols, I have replaced them with suitable alphabets.

The basic equation we are trying to solve is to get the numbers :
x & y and u & v and N that satisfy the Diophantine equation :
x^3 + y^3 = u^3 + v^3 = N
where the four integers x, y, u, v have no common factor.

For eg, the "lowest" Diophantine Number N of Ramanujam is :
1729 = 9^3 + 10^3 = 1^3 + 12^3

Another example is :
(-107766)^3 + (-634932)^3 = (-2013055)^3 + 1991671^3
= -257217167508536664 (Calculator) { -2.572171675085367e+017 (Matlab) }
ie, 2013055^3 = 107766^3 + 634932^3 + 1991671^3

I wish to thank Prof John Butcher for his article which triggered and enabled me to write this MATLAB code for "generating" Ramanujam's Diophantine Equation Numbers. I hope that this programme will be useful to many the world over. Time permitting, I may be improving on this programme to make it suitable for generating a series of Diophantine Numbers automatically. But the base groundwork is already laid now, and we need only to build further.

Q1a : I am curious to know why Prof John Butcher has said that gcd (a, b) and gcd (c, d) should be 1.
In the case of his own example : 9^3 + 15^3 = 2^3 + 16^3 = 4104,
intermediate calculations give a = 12, b = 3, c = 9, d = 7.
See Usage Eg Case 4 below. Obviously, gcd (a, b) = 3 (not 1).
But, gcd (x, y, u, v) = 1 (var name used in my programme is G_Gxy_Guv.)

Also, in his second example : 33^3 + 15^3 = 2^3 + 34^3 = 39312,
calculations give a = 3, b = 9, c = 9, d = 8. See Usage Eg Case 5 below.
Obviously, gcd (a, b) = 3 (not 1). But, gcd (x, y, u, v) = 1.

Similarly, in his 3rd example too, gcd (a, b) = 3 (not 1) !

In many of my own examples with randomly chosen values, I have mixed results:
In Usage Eg 2, gcd (c, d) = 9 (not 1), but, gcd (x, y, u, v) = 1.
In Usage Eg 3, gcd (a, b) = 3 (not 1), but, gcd (x, y, u, v) = 3 (not 1) !
Egs 7 and 8 are similar to Eg 2 ; Egs 9, 10, and 11 are similar to Eg 3.

Q1b) Therefore, I would like to know the condition or constraint, which when fulfilled, will ensure that we will certainly obtain
x, y, u, v such that gcd (x, y, u, v) = 1

Q2 : I would also like to know how the article's Theorem 1 is used in deriving the formulae for Diophantine Numbers.

It says : ... "although beneath the surface" ...

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Now added in the suite :
find_x_y__p_1_mod_6.m : This function finds x and y pairs of numbers such that x2 + 3y2 = p where mod(p, 6) = 1