# Ramanujam Diophantine Equation 1.0

OS : Windows / Linux / Mac OS / BSD / Solaris

Script Licensing : Freeware

Created : Aug 27, 2007

Downloads : 5

Thank you for voting...

## This programme is about finding many examples of ...

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, butcher@math. auckland. ac. nz.

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

where the four integers x, y, u, v have no common factor.

For eg,

= -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,

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" . . .

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 = Nwhere 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" . . .

**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**• MATLAB Release: R13**

**Demands:****Ramanujam Diophantine Equation 1.0 scripting tags:**matlab mathematics, equation, ramanujam, diophantine equation, programme, ramanujam diophantine, gcd.

**What is new in Ramanujam Diophantine Equation 1.0 software script?**- Unable to find Ramanujam Diophantine Equation 1.0 news.

**What is improvements are expecting?**Newly-made Ramanujam Diophantine Equation 1.1 will be downloaded from here. You may download directly. Please write the reviews of the Ramanujam Diophantine Equation. License limitations are unspecified.