Equal sums of quartics (In context with the Richmond equation) (

Consider the below mentioned equation: ) In section (A) we consider solution’s with the condition on the coefficient’s of equation (1). Namely the product ( ) square. In section (B) we consider the coefficients of equation (1), with the product of coefficient’s (abcd) not equal to a square. Historically equation (1) has been studied by Ajai Choudhry, A. Bremner, M.Ulas (ref. 5) in 2014. Also Richmond (ref. 1 & 2) has done some ground work in 1944 & 1948. This paper has gone a step further, by finding many parametric solutions & new small numerical solutions by the use of unique Identities. The identities are unique, because they are of mixed powers(combination of quartic & quadratic variables) which are then converted to only degree four identities. As an added bonus in section (B), we came up with a few quartic ( ) numerical solutions for ( ) by elliptical mean’s. A table of numerical solutions for the(4-1-n)equation arrived at by brute force computer search is also given (ref # 7).

We consider below equation: Where, in equation (1), ( a, b, c, d)=( p, q,-p,-q) To obtain a rational solution of equation (2) for t, we have to find the rational solution of (3).
( ) Thus we obtain a parametric solution.
Furthermore, we can find other parametric solution by using this method.
Hence by repeating this process, we can obtain infinitely many integer solutions for equation (1).
Example: If 'm' is written as (p/q) the above can be re-written as below: ( ) ( ) m,n are arbitrary.
By parameterizing the first equation and substituting the result to second equation, then we obtain quartic equation below to make (w= square). Since, (t, z) ( ) we get ( ) ( ) so we parameterize for (t, z) We get: Hence we get: This quartic equation (4) is bi-rationally equivalent to an elliptic curve below.
( ) This point P is of infinite order, and the multiples nP, n = 2, 3, ...give infinitely many points.
This quartic equation (4) above has infinitely many parametric solutions one of which is shown below.
Hence we can obtain infinitely many integer solutions for equation (1).
has infinitely many integer solutions. d, m are arbitrary.
So, we look for the integer solutions By parameterizing the first equation and substituting the result to second equation, then we obtain quartic equation (4) below. (4) is bi-rationally equivalent to an elliptic curve below.

This quartic equation
This point P is of infinite order, and the multiples nP, n = 2,3, ... give infinitely many points.
This quartic equation has infinitely many parametric solutions below.
Hence we can obtain infinitely many integer solutions for equation (1).
We show above Diophantine equation has infinitely many integer solutions. d,m are arbitrary.
We use an identity, By parameterizing the first equation and substituting the result to second equation, then we obtain quartic equation below.
This quartic equation is bi-rationally equivalent to an elliptic curve below.
This point P is of infinite order, and the multiples nP, n = 2,3, ...give infinitely many points.
This quartic equation has infinitely many parametric solutions below. n=2, Hence we can obtain infinitely many integer solutions for equation (1).
Example: Case A:

Substitution above values in equation (1) we get:
Above is equivalent to: Hence we get: Above equation (2) is parameterized as: And as, & we get the numerical solution: Therefore we take, Thus we have, Note: Reader's may be interested in the table of result's below. These were arrived at by elliptical method. We have not attempted to parameterize the family of equations below, but others can make an attempt. Since knowing the numerical solution is helpful towards parameterization.  w   19  5  13  2  23  9  1  26  73  23  39  22  13  3  11  2  17  7  1  22  9  131  157  92  1  31  41  58  17  7  15 10
Hence we can obtain infinitely many integer solutions for equation (1).
Numerical example: From above we deduce: There are more (4-1-n) numerical solution's as derived by elliptical method for ( n < 50) & is shown below:  We use an identity: By parameterizing the second equation and substituting the result to first equation, then we obtain quartic equation below.
This quartic equation is bi-rationally equivalent to an elliptic curve below.
Hence we get 2P(X,Y), This point P is of infinite order, and the multiples mP, m = 2,3, ...give infinitely many points.
This quartic equation has infinitely many parametric solutions below.
Hence we can obtain infinitely many integer solutions for equation (1).

Example:
Numerical example: We show Diophantine equation ( ) has infinitely many integer solutions for b is arbitrary.
This quartic equation is bi-rationally equivalent to an elliptic curve below.
The corresponding point is ( This point P is of infinite order, and the multiples mP, m = 2,3, ...give infinitely many points.
This quartic equation has infinitely many parametric solutions below. .
Hence we can obtain infinitely many integer solutions for equation (1). Example: ) has infinitely many integer solutions. d is arbitrary.
So, we look for the integer solutions This quartic equation is birationally equivalent to an elliptic curve below.
This point P is of infinite order, and the multiples nP, n = 2,3, ...give infinitely many points.
This quartic equation has infinitely many parametric solutions below. For n=2 Hence we can obtain infinitely many integer solutions for equation (1). Example: Re: d=1, Is equivalent to (4-1-25) equation: We show Diophantine equation above has infinitely many integer solutions. 'm' is arbitrary.
We use an identity: So, we look for the integer solutions This quartic equation is bi-rationally equivalent to an elliptic curve below.
This quartic equation has infinitely many parametric solutions below. For n=2, ( ) Hence we can obtain infinitely many integer solutions for equation (1). Example: For m=2, multiplying throughout by two we get: After removing common factor's we get: Also see Table no. 4 at the end of this paper for results arrived at by brute force for equation ( ). For (abcd) not equal to a square & k=(a+b+c+d) for ( k < 30 ).