Universal Journal of Applied Mathematics
Vol. 8(2): 13 - 29, 2020, DOI:10.13189/ ujam.2020.080 2 0

http://www.hrpub.org

Equal Sums of Quartics (In Context with the RichmondEquation) ()Seiji Tomita1, Oliver Couto2
1

Tokyo Computer Company, Tokyo, Japan
University of Waterloo, Waterloo, Ontario, Canada

2

Email: (1) fermat@m15.alpha-net.ne.jp
Email: (2): samson@celebrating-mathematics.com
Received ; October 12, 2020;

Copyright©2020 by authors, all rights reserved. Authors agree that this article remains permanently open access under
the terms of the Creative Commons Attribution License 4.0 International License

Abstract: Consider the below mentioned Equation:

In section (1) we
consider solution’s with the condition on the coefficient’s of equation[1]. Namely the productsquare. In
section [2] 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 parametricsolutions & 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. Asan added bonus in section [B], we came up with a few quartic
numerical solutions for) byelliptical 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].

Keywords Quartic Equation, Diophantine Equation, Diagonal Quartic Surfaces, Number Theory

1. Section (1)=0[Condition:The productsquare](1)Case (A)We show Diophantine Equation,(1)
has infinitely many integer solutions. m, n are arbitrary.
We consider below Equation:
Where, in Equation (1), ( a, b, c, d)=( p, q,-p,-q)(2)
To obtain a rational solution of Equation (2) for t, we have to find the rational solution of (3).
(3)
Let consider Equation (3) is a quadratic Equation for p.

Thus we obtain a parametric solution.

Where, m,n are arbitrary.
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:()Case (B):We show Diophantine Equation:=(1a)m is arbitrary.
Since in (1a) above, ( a, b, c, dIf ‘m’ is written as (p/q) the above can be re-written as below:] the product (abcd) becomes a square=[.

m,n are arbitrary.
(1)
(2)
()

{}

(3)

By parameterizing the first Equation and substituting the result to second Equation, then we obtain quartic Equation
below to make (w= square).
Since, (t, zwe getso we parameterize for (t, z)
We get:Hence we get:(3a)In-order to make the coefficient ofa square we take)(4)

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.

n=3

etc.
Hence we can obtain infinitely many integer solutions for Equation (1).
Example: m=2,Case (C):We show Diophantine Equation,
has infinitely many integer solutions.
d, m are arbitrary.
(1)
We use an identity(2)

So, we look for the integer solutions{}(3)

By parameterizing the first Equation and substituting the result to second Equation, then we obtain quartic Equation (4)below.

(4)
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 has infinitely many parametric solutions below.

etc.
Hence we can obtain infinitely many integer solutions for Equation (1).
Example:Case (D)We show above Diophantine Equation has infinitely many integer solutions. d,m are arbitrary.
(1)
We use an identity,
(())

(2)

(3)

By parameterizing the first Equation and substituting the result to second Equation, then we obtain quartic Equation
below.

(4)
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,etc.
Hence we can obtain infinitely many integer solutions for Equation (1).
Example:Or

Equation of the type:(1)
where:
Note: We have attempted to parameterize a similar family of Equation related to Equation (1) at the top of the paper.
Difference being Equation above is mixed powers of degree four & two. Others are free to attempt to make the two squarecomponents a fourth power. Thus making Equation (2) eligible for the Equation (1) group.
Case A:=0Let,
Substitution above values in Equation (1) we get:Above is equivalent to:Since,), let:(2)

Hence we get:

Therefore we get:,&(2)

Above Equation (2) is parameterized as:Hence we get:Since,
we get,And as,we get,Summarizing,for,)& we get the numerical solution:
Additional Example:
Case B:(1)

Since we have,Hence,,
Therefore we take,Thus we have,
Taking,

we get,For,) we get,Note: Reader’s may be interested in the table of result’s below. These were arrived at by elliptical method. We have notattempted to parameterize the family of Equations below, but others can make an attempt. Since knowing the numericalsolution is helpful towards parameterization.
Table 1.

(below for numerical solutions):

Equation type:product (abcd) = □ (square),& Condition, (a+b)=(c+16d)

xY

Zw

195132

23

9

1

26

73

23

39

22133112

177

1229131

157

92

13141

58

17715102. Section (2)=0(1)[Condition: Product of coefficient’s (abcd) not equal to a square]
Case (A)Introduction:
Existence of solution for Diophantine Equation
are known if (abcd) is square number.
So, we are curious about whether above Equation has a solution or not if abcd is not a square number. In particular,when does this Equation have infinitely many integer solutions Since Richmond (ref # 2) conjectured that, if there is one
rational solution there are infinite more.
We show Diophantine Equation:
For (p,q,r,s)=(7,7,1,8)
has infinitely many integer solutions (numerical method).
Multiplying throughout by two the above Equation we get:
Which is a (4-1-30) quartic Equation:(1)
We use an identity
()

(2)

{}

(3)

By parameterizing the second Equation and substituting the result to first Equation, then we obtain quartic Equation
below.
(4)
This quartic Equation is bi-rationally equivalent to an elliptic curve below.

Hence we can obtain infinitely many integer solutions for Equation (1).
Numerical example:
From above we deduce:
(1)
Let :
For (p,q,r)=(14,14,2)
Let: n=( p+q+r )Above Equation (1) is a (4-1-n) quartic Equation. For n=(14+14+2)=30 which we write as (4-1-n)=(4-1-30) Equation.
There are more (4-1-n) numerical solution’s as derived by elliptical method for (n < 50) & is shown below:(4-1-30) eqn.
(4-1-16) eqn.
(4-1-40 eqn.
4-1-36) eqn.
(4-1-16)
Note: For more (4-1-n) Equation numerical solutions interested reader can see table (3) at the end of this paper, (ref #12).
Parametric method in-lieu of the above elliptical numerical method (eqn. (4):We show Diophantine Equation
has infinitely many integer solutions.
(1)
We use an identity:
(2)

{}

(3)

By parameterizing the second Equation and substituting the result to first Equation, then we obtain quartic Equation
below.
(4)
This quartic Equation is bi-rationally equivalent to an elliptic curve below.

The corresponding point is.

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.

etc.
Hence we can obtain infinitely many integer solutions for Equation (1).
Example:Numerical example:

Case (B):We show Diophantine Equationb is arbitrary.

has infinitely many integer solutions for(1)

We use an identity(2)

So, we look for the integer solutions,(3)
By parameterizing the second Equation and substituting the result to first Equation, then we obtain quartic Equation
below.
(4)
This quartic Equation is bi-rationally equivalent to an elliptic curve below.

The corresponding point isThis 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.
.

etc.
Hence we can obtain infinitely many integer solutions for Equation (1).
Example:Case (C)We show Diophantine Equation,integer solutions.
d is arbitrary.

has infinitely many

=()

(1)
(2)

()

So, we look for the integer solutions
{()}

(3)

By parameterizing the second Equation and substituting the result to first Equation, then we obtain quartic Equation
below.
(4)
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

For n=3,etc.
Hence we can obtain infinitely many integer solutions for Equation (1).
Example:Re: d=1, Is equivalent to (4-1-25) Equation:
Case (D)We show Diophantine Equation above has infinitely many integer solutions.
‘m’ is arbitrary.
(1)
We use an identity:(2)

So, we look for the integer solutions(3)
By parameterizing the second Equation and substituting the result to first Equation, then we obtain quartic Equation
below.
then,(4)This quartic Equation is bi-rationally equivalent to an elliptic curve below.

The corresponding point is P(X,Y)=().
()

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,For n=3,etc.
Hence we can obtain infinitely many integer solutions for Equation (1).
Example:For m=2, multiplying throughout by two we get:() which is a (4-1-79) quartic Equation.
Example:(1)
Note: We have attempted to parameterize a similar family of Equation related to Equation (1) at the top of the paper.
Difference being Equation above is mixed powers of degree four & two. Others are free to attempt to make the two squarecomponents a fourth power. Thus making Equation (2) below eligible for the Equation (1) group above.
(2)(pqrs) not a square,Hence Equation (1) becomes:

(3)Equation (3) has solution:
For t=6, 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 ).
(See Table (2) below), for numerical solutions:
Note: Reader’s may be interested in the table of result’s below. These were arrived at by elliptical method. We have notattempted to parameterize the family of Equations below, but others can make an attempt. Since knowing the numericalsolution is helpful towards parameterization.
Table (2)=0=XYZW360

4345737151693

41613

38547

81294

3689

16892

17818

46819

22151728200

529

460

721

412972112161724984

113

91719148209

46228

81169

14397

40473757

1711732X

Y

Z=

2356

47

93=

30

11921352

235

917

894

=

23

75=

13337

47

437

1=

=Equation: (table 3a)

=
=

=

=

4391

3769

2361

5481

1758

13587

3111103720

56

83

25

8

193172355

239

329

940

30

41=

142

7=

112617=4854737952

475

2073

1123

2649559249

[See Table (3a), below for (a,b,c,d) arbitary],

Table (3b).

[5]
[6]
[7]
[8]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[43]
[44]
[45]
[46]
[47]
[48]
[49]

a3
4
355
78
1
978
99910
913
14
13
1110111113
13
14
15
17
18
18
1731
2625
2218
2230
33
24
1919432322b1132433147577777678899101211111010991557817141911819141731319C111111141131233532367774667678415619122112101128x102694
20104351026173111331711812252021225
1021231214174521431y1712240106061124381151173110102222925
7815
1172
753455132416
1033853z
30
2569
17
3530
1761
111101024111915359
55919
7151233
32524425520
21
6645
15w
31
329143
3179
93935
75
225122856182442598
44131743113551549679956983125
131149109

Table 4. (See below)
Serial nos.

abcdxyzw

1

6-3-3

123

14760

35-2-2242

51421-122354542-21233744

654-4120173140

77-3-366571852-231872310986-611051181072-2321321185-5310913101275-3342511373-36313302514112-235910

11584-4

726731816

107-2

115114755

17108-87435418133-24518219124-25156220

87-38124321115-275912422126-59589323

98-28231072498-395411725

126-310416326

137-410111151027

117-1

102513728

15

1-11323191029129-3

1157103

Acknowledgment:
The authors are thank-full tomathematician Dr. Ajai Choudhry for hiscomments about this paper
Conclusion:
Re: Equation, (= 0 ----(1),while in section (A) we noticed, that when we imposed
condition’s on Equation (1) like (abcd) = □ &((a+b)=(c+16d)), we got small numerical solutions for thevariable’s (X,Y,Z,W) [see table (1)]. But in section (B) intable (2a & 2b) when we used different numerical values ofcoefficients (a,b,c,d) for Equation (1) a clear relationshiphas not emerged between the coefficient’s (a,b,c,d). For
example,in,some,cases,we,took:(a,b,c,d)=(23,23,2,25)=(1,36,27,4)=(49,49,8,1)=(3,75,5,8).
The interested reader may try to find a pattern in the above(a,b,c,d), which could facilitate discovering more similar
Identities of the type given for case (1) in section (B)shown below for example:.

(1998), (1127-1128).

REFERENCES[1] A Note upon an Arithmetical Property of Quartic Surfaces
Herbert W. Richmond, vol. s1-23, issue 1, Jan 1948
[2] H. W. Richmond, on the Diophantine Equation,; The productbeing asquare number, J. London Math. Soc.19 (1944), (193-194).
[3] Constructions of diagonal quartic and sextic surfaces with
infinitely many rational points. Preprint, Arxiv 1402, 19Feb. 2014, Authors: Ajai Choudhry, Andrew Bremner,Maciej Ulas
[4] Seiji Tomita, webpage,, web site on “Computation number theory” Ref.
Article # 391,397 & 398. Web page includes listing of
published math papers by author.
[5] Ajai Choudhry, the Diophantine Equation
Indian Journal of Pure and Applied Mathematics,
22 (1991), (9-11).
[6] OliverCouto,

webpage,, web site on “Number theorymathematics” Ref. (articles by O. Couto), art # 12. Web pagealso has reference to published math papers by author.

[7] Ajai Choudhry, on the Diophantine Equation,
Indian Journal of Pure and Applied Mathematics,(26 (1995), (1057-1061).
[8] J.H. Silverman: Rational points on elliptic curves, arithmeticof elliptic curves,1992[9] Ajai Choudhry, The Diophantine Equation,[10], Indian Journal of Pure Applied Mathematics, 29[11] Jarosław Wróblewski & A. Choudhry, an Ancient
Diophantine Equation with applications to Numerical Curiosand Geometric Series.
[12] Ajai Choudhry, on a quartic Diophantine Equation, The
Mathematics Student, 70 (2001), (181-183).
[13] Table (3) of numerical solution’s for the (4-1-n) Equation,
provided by Seiji Tomita via email[14] Noam Elkies, on eqn.
51 (1988), 825-835.

, Math. Comp.

[15] Crenome’s & Stein-Watkins tables for elliptic curves[16] Computer software: Pari, Maple & Magma[17] Elliptic curves tables: http://www. Imfdb.org[18] Tito Piezas-online book Collection of Algebraic identities,(PartSeven,Fourthpowers)[19] R. D. Carmichael, on certain Diophantine Equations having
multiple Parameteric solutions, Amer. Math. Monthly 20(1913), 304-307.
[20] A-S. Elsenhans, Rational points on diagonal quartic surfaces,
Math. Comp. 81 (2012), 481-492.
[21] A. Logan, D. McKinnon, R. Van Luijk, Density of rational
points on diagonal quartic surfaces, Algebra and Number
Theory 4 (2010), (1-20).
[22] A-S. Elsenhans and J. Jahnel, The diophantine Equation
Math. Comp. 75 (2005),(935-940).
[23] P. Swinnerton-Dyer, Arithmetic of diagonal quartic surfaces,
Proc. London Math. Soc. 80 (2000), pages (513-544).