*Vol. 10, 2017, 57-65 *

*ISSN: 2349-0632 (P), 2349-0640 (online) *
*Published 11 December 2017 *

*www.researchmathsci.org*

*DOI: http://dx.doi.org/10.22457/jmi.v10a8 *

57

Journal of

**Observation on the Cubic Diophantine Equation with **

**Four Unknowns (x**

**Four Unknowns (x**

**3**

**+y**

**+y**

**3**

**)+(x+y)(x+y+1)=zw**

**)+(x+y)(x+y+1)=zw**

**2**

* T. Priyadharshini1 and S. Mallika2 *
1

Department of Mathematics, Shrimati Indira Gandhi College, Trichy-2 Tamilnadu INDIA, e.mail: jpdharshini16@gmail.com

2

Department of Mathematics, Shrimati Indira Gandhi College, Trichy-2 Tamilnadu INDIA, e.mail: msmallika65@gmail.com

*Received 1 November 2017; accepted 06 December 2017 *

**Abstract. The non-homogeneous cubic Diophantine equation represented by**

### (

3 3### )

### (

### )(

### )

2### 1

*zw*

*y*

*x*

*y*

*x*

*y*

*x*

### +

### +

### +

### +

### +

### =

is analyzed for its non-zero distinct integer solutions. A few interesting relations between the solutions and Polygonal numbers,

**Pyramidal numbers are also presented.****Keywords: Integer solutions, non-homogeneous cubic, Polygonal numbers, Pyramidal ****numbers. **

**AMS Mathematics Subject Classification (2010): 11D25 **

**1. Introduction **

Integral solution for the Homogeneous or Non- homogeneous Diophantine cubic equation is an interesting concept as it can be seen from [1,2,3]. In [4−11] a few special cases of cubic Diophantine equation with four unknowns are studied. In this communication we present the integral solutions of an interesting cubic equations with four unknowns

2 3

3

### )

### 1

### )(

### (

### )

### (

*x*

### +

*y*

### +

*x*

### +

*y*

*x*

### +

*y*

### +

### =

*zw*

. A few remarkable relations between the solutions
are also presented.
**2. Notations **

=
*n*
*m*

*t* _{,} nth term of a regular polygon with m sides

### (

### )(

### )

_{}

− −

+ =

2 2 1

1 *n* *m*

*n*

### Pr

*n*= Pronic number of rank n.

=*n*(*n*+1)

*n*

*S*

= Star number of rank n.
58

6 ,

*n*

*CP* = Centered Pyramidal Number of rank n with sides m

3
*n*

=

**3. Method of analysis **

The cubic Diophantine equation with four unknowns to be solved for its non-zero distinct integral solution is

### (

3 3### )

2### )

### 1

### )(

### (

*x*

*y*

*x*

*y*

*zw*

*y*

*x*

### +

### +

### +

### +

### +

### =

_{ }

_{ (1) }

The substuition of the linear transformations

*v*

*u*

*x*

### =

### +

,*y*=

*u*−

*v*

### (

*u*≠

*v*≠0

### )

(2) in (1) leads to2 2 2

3*v* *w*

*U* + = (3)
where *U* =*u*+1 (4)
Different patterns of solutions of (1) are presented below

**3.1. PATTERN-1 **

Assume

2 2

*3b*

*a*

*w*= + ** ** ** (5) **

*where a and b are non-zero distinct integers. *
Substituting (5) in (3), we get

2 2 2 2 2

### )

### 3

### (

### 3

*v*

*a*

*b*

*U*

### +

### =

### +

(6)Employing the method of factorization, we get

2 2

) 3 (

) 3 (

) 3 )(

3

(*U* +*i* *v* *U* −*i* *v* = *a*+*i* *b* *a*−*i* *b*

Equating the positive and negative factors, we get )

3

(*U* +*i* *v* = (*a*+*i* 3*b*)2
)

3

(*U* −*i* *v* = (*a*−*i* 3*b*)2
Equating the real and imaginary part, we get

2 2

*3b*

*a*

*U* = − (7)

*ab*

*v*=2 (8)

In view of (4)

1 3 2

2 − −

=*a* *b*

*u* (9)

Substituting (8) and (9) in (2), we get the distinct non-zero integer solutions of (1) are given below

### ( )

### ,

### =

2### −

### 3

2### +

### 2

### −

### 1

### =

*x*

*a*

*b*

*a*

*b*

*ab*

*x*

### ( )

### ,

### =

2### −

### 3

2### −

### 2

### −

### 1

### =

*y*

*a*

*b*

*a*

*b*

*ab*

*y*

### 2

### 6

### 2

### )

### ,

### (

### =

2### −

2### −

### =

*z*

*a*

*b*

*a*

*b*

*z*

*(x*3*+y*3*)+(x+y)(x+y+1)=zw*2

59
1. *x*

### ( ) ( )

*a*,1 +

*y*

*a*,1 +

*z*(

*a*,1)−4

*t*

_{4}

_{,}

*+16=0 2.*

_{a}*y*

### (

*a*,2

*a*−1

### )

+*t*32,

*a*+4=0

3. *x*

### ( ) ( ) ( )

*a*,

*a*+

*ya*,

*a*+

*z*

*a*,

*a*+

*w*(

*a*,

*a*)+4

*t*

_{4}

_{,}

*+4=0 4.*

_{a}*z*

### ( ) ( )

*a*,1 −

*wa*,1 −

*t*4,

*a*+11=0

**3.2. PATTERN-2 **

Write (5) as *a*2 +3*b*2 =*w**1 (10)
Now write 1 as

## (

## )(

## )

4 3 1 3 1

1= +*i* −*i* (11)
Using (10) and (11) in (3) and employing the method of factorization, the above
equation (3) is written as

## (

## )(

## ) (

## )(

## )(

## ) (

2## )

23 3

4 3 1 3 1 3

3*v* *U* *i* *v* *i* *i* *a* *i* *b* *a* *i* *b*
*i*

*U*+ − = + − + −

Equating the positive and negative factors, we get

## (

_{)(}

_{)}

2
3 2

3 1

3*v* *i* *a* *i* *b*
*i*

*U*+ = + + (12)

## (

_{)(}

_{)}

2
3 2

3 1

3*v* *i* *a* *i* *b*
*i*

*U*− = − −

(13)

Equating the real and imaginary parts, we get

### (

*a*

*b*

*ab*

### )

*U* 3 6

2

1 2 − 2 −

=

(14)

### (

*a*

*b*

*ab*

### )

*v* 3 2

2

1 2 − 2 +

=

(15)

* Replacing a,b by 2A,2B respectively, we get *

*AB*
*B*

*A*

*U* =2 2 −6 2 −12 _{ (16)}

*AB*
*B*

*A*

*v*=2 2 −6 2 +4

(17) In view of (4),

1 12 6

2 2 − 2 − −

= *A* *B* *AB*

*u* _{ (18) }

*Substituting the above values ofu and v in (2), we get *

### ( )

### ,

### =

### 4

2### −

### 12

2### −

### 8

### −

### 1

### =

*x*

*A*

*B*

*A*

*B*

*AB*

*x*

_{ }

### ( )

, =−16 −160

### 2

### 24

### 12

### 4

### )

### ,

### (

### =

### −

### −

### −

### =

*z*

*A*

*B*

*A*

*B*

*AB*

*z*

2 2

### 12

### 4

### )

### ,

### (

*A*

*B*

*A*

*B*

*w*

*w*

### =

### =

### +

represents the non-zero distinct integral solutions of (1)

**PROPERTIES **

**1.**

### 6

### [

*w*

### ( )

### 1

### ,

### 1

### ]

### =

### 6

### ( )

### 4

2**is a Nasty Number**

2. *y*

### (

*n*(

*n*−1,1

### )

+*Sn*+

*t*22,

*n*≡0

### (

mod*n*)

### )

3. *w*

### ( ) ( )

1,*B*−

*z*1,

*B*−

*t*

_{50}

_{,}

*−2≡1*

_{B}### (

mod2### )

4. *w*

### (

*A*,

*A*+1

### ) (

−*y*

*A*,

*A*+1

### )

−16*PRA*−

*t*34,

*A*−13≡0

### (

mod13### )

**NOTE: **

We note that, if we replace *a,b*by 2*A*+1,2*B*+1 respectively in (14) and (15) we get
the different set of distinct non-zero integer solutions of (1) given by

### ( )

### ,

### =

### 4

2### −

### 12

2### −

### 16

### −

### 8

### −

### 5

### =

*x*

*A*

*B*

*A*

*B*

*B*

*AB*

*x*

### ( )

, =−8 −8 −16 −5= *y* *A* *B* *A* *B* *AB*

*y*

### 10

### 24

### 24

### 8

### 12

### 4

### )

### ,

### (

### =

2### −

2### −

### −

### −

### −

### =

*z*

*A*

*B*

*A*

*B*

*A*

*B*

*AB*

*z*

### 4

### 12

### 4

### 12

### 4

### )

### ,

### (

### =

2### +

2### +

### +

### +

### =

*w*

*A*

*B*

*A*

*B*

*A*

*B*

*w*

**PROPERTIES **

1.

### 6

### [

*w*

### ( )

### 1

### ,

### 1

### ]

### =

### 6

### ( )

### 6

2 is a Nasty Number 2.*w*(

*A*,1)−

*y*

### ( )

*A*,1 −

*t*6,

*A*−41≡1

### (

mod3)### )

3.*x*(

*A*,1)−

*w*

### ( )

*A*,1 +61≡0

### (

mod3### )

4. *x*(*A*,1)+ *z*(*A*,1)+*w*

### ( )

*A*,1 −

*t*

_{26}

_{,}

*+51≡0*

_{A}### (

mod5### )

**3.3. PATTERN-3 **

We can also write 1 as,

## (

## )(

## )

49

3 4 1 3 4 1

1= +*i* −*i*

(19)

Using (5) and (19) in (3) and employing the method of factorization, we get

## (

## )(

## ) (

## )(

## )(

## ) (

2## )

23 3

49

3 4 1 3 4 1 3

3*v* *U* *i* *v* *i* *i* *a* *i* *b* *a* *i* *b*

*i*

*U*+ − = + − + −

Equating the positive and negative factors, we have

## (

_{)(}

_{)}

2
3 7

3 4 1

3*v* *i* *a* *i* *b*

*i*

*(x*3*+y*3*)+(x+y)(x+y+1)=zw*2

61

## (

_{)(}

_{)}

2
3 7

3 4 1

3*v* *i* *a* *i* *b*

*i*

*U*− = − −

Equating the real and imaginary parts ,

### (

*a*

*b*

*ab*

### )

*U* 3 24

7

1 2 − 2 −

=

(20)

### (

*a*

*b*

*ab*

### )

*v* 4 12 2

7

1 2 _{−} 2 _{+}

=

(21)

*Replacing a,b by 7A ,7B respectively, we get *

*AB*
*B*

*A*

*U* =7 2 −21 2 −168 _{ (22)}

*AB*
*B*

*A*

*v*=28 2−84 2+14

(23) In view of (4)

### 1

### 168

### 21

### 7

2### −

2### −

### −

### =

*A*

*B*

*AB*

*u*

_{ (24)}

Substituting (23) and (24) in (2), we get

### ( )

### ,

### =

### 35

2### −

### 105

2### −

### 154

### −

### 1

### =

*x*

*A*

*B*

*A*

*B*

*AB*

*x*

### ( )

### ,

### =

### −

### 21

2### +

### 63

2### −

### 182

### −

### 1

### =

*y*

*A*

*B*

*A*

*B*

*AB*

*y*

### 2

### 336

### 42

### 14

### )

### ,

### (

### =

2### −

2### −

### −

### =

*z*

*A*

*B*

*A*

*B*

*AB*

*z*

2 2

### 147

### 49

### )

### ,

### (

*A*

*B*

*A*

*B*

*w*

*w*

### =

### =

### +

represents the non-zero distinct integral solutions of (1).

**PROPERTIES **

1.

### 6

### [

*w*

### ( )

### 1

### ,

### 1

### ]

### =

### 6

### ( )

### 14

2 is a Nasty Number2. *z*(*A*,1)+ *y*

### ( )

*A*,1 +

*w*(

*A*,1)−

*t*86,

*A*−165≡0

### (

mod159### )

3. *z*(*AA*)+364*t*4,*A* +2=0

4. *x*

### ( )

*A*,1 −

*t*

_{72}

_{,}

*+106≡0*

_{A}### (

mod2### )

**3.4. PATTERN-4 **

(3) is written in the form of ratio as

0 ,

3 = − = ≠

+

*n*
*n*
*m*
*U*
*w*

*v*
*v*

*U*
*w*

(25)

which is equivalent to the system of equations, 0

3 =

−

+*nU* *mv*

*nw* (26)
0

=
−
−*mU* *nv*

*mw* (27)

62
3*m* *n*

*U* = −

*mn*
*v*=2

2 2

### 3

*m*

*n*

*w*

### =

### +

In view of (4), 1 3 2 − 2 −

= *m* *n*

*u*

Substituting the values of *u* and *v* in (2), we get the non zero distinct integer solutions
of (1) is given by

### ( )

### ,

### =

### 3

2### −

2### +

### 2

### −

### 1

### =

*x*

*m*

*n*

*m*

*n*

*mn*

*x*

### ( )

### ,

### =

### 3

2### −

2### −

### 2

### −

### 1

### =

*y*

*m*

*n*

*m*

*n*

*mn*

*y*

### 2

### 2

### 6

### )

### ,

### (

### =

2### −

2### −

### =

*z*

*m*

*n*

*m*

*n*

*z*

2 2

### 3

### )

### ,

### (

*m*

*n*

*m*

*n*

*w*

*w*

### =

### =

### +

**PROPERTIES **

1.

*x*

### (

*m*

### ,

*m*

### +

### 1

### ) (

### −

*y*

*m*

### ,

*m*

### +

### 1

### )

### −

### 4

*PR*

_{m}### =

### 0

2. *x*

### (

*m*,

*m*+1

### ) (

+*ym*,

*m*+1

### )

−*t*12,

*m*+

*PRm*+4≡0

### (

mod*m*

### )

3. *x*

### ( ) ( )

*m*,1 +

*y*

*m*,1 +

*z*(

*m*,1)+

*w*(

*m*,1)−15

*t*4,

*m*+7=0

4. *x*(*m*,*m*+1)+ *y*

### (

*m*,

*m*+1

### )

+*z*(

*m*,

*m*+1)−

*t*

_{20}

_{,}

*+*

_{m}*PR*+8≡0

_{m}### (

mod*m*

### )

**3.5. PATTERN-5 **

One may write (3) in the form of ratio as

0 ,

3 _{=} _{≠}

− = +

*n*
*n*
*m*
*U*
*w*

*v*
*v*

*U*
*w*

(28)

which is equivalent to the system of equations 0

=
−
+*nU* *mv*

*nw* (29)
0

3 =

−

−*mU* *nv*

*mw* (30)

Applying the method of cross multiplication for solving (29) and (30), we have

2 2

*3n*

*m*

*U*

### =

### +

*mn*
*v*=2

2 2

*3n*

*m*
*w*= +

In view of (4),

### 1

### 3

22

### −

### −

### =

*m*

*n*

*u*

Substituting the values of

*u*

and *v*

in (2), we have
### ( )

### ,

### =

2### −

### 3

2### +

### 2

### −

### 1

### =

*x*

*m*

*n*

*m*

*n*

*mn*

*x*

### ( )

### ,

### =

2### −

### 3

2### −

### 2

### −

### 1

### =

*y*

*m*

*n*

*m*

*n*

*mn*

*y*

### 2

### 6

### 2

### )

### ,

### (

### =

2### −

2### −

### =

*z*

*m*

*n*

*m*

*n*

*z*

2 2

### 3

### )

### ,

### (

*m*

*n*

*n*

*m*

*w*

*(x*3*+y*3*)+(x+y)(x+y+1)=zw*2

63

represents the distinct non zero integral solutions of (1)

**PROPERTIES **

1.

*x*

### (

*m*

### ,

### 1

### )

### −

*w*

### (

*m*

### ,

### 1

### )

### +

### 7

### ≡

### 0

### (mod

### 2

### )

2. *z*(*m*,*m*+1)− *y*

### (

*m*,

*m*+1

### )

−2*PRm*+

*t*6,

*m*+4≡0

### (

mod7### )

3.*x*

### (

1,*n*+1

### ) (

+*y*1,

*n*+1

### ) (

+*z*1,

*n*+1

### )

+*w*(1,

*n*+1)−

*t*

_{12}

_{,}

*+36=0 4.*

_{m}*x*

### ( ) ( )

*m*,1 +

*y*

*m*,1 +

*w*(

*m*,1)−3

*t*

_{4}

_{,}

*+5=0*

_{m}**3.6. PATTERN-6 **

Introduction of the linear transformations
*T*

*X*

*U* = +3 *v*= *X* −*T* _{ }
(31)
in (3) gives,

2 2 2

### 12

### 4

*X*

### +

*T*

### =

*w*

_{ }(32) which is satisfied by,

2 2

6
2*a* *b*
*X* = −

*ab*
*T* =4

*b*

*a*

*w*

### =

### 4

2### +

### 12

*Substituting the values of X,T in (31), we get *

*ab*

*b*

*a*

*U*

### =

### 2

2### −

### 6

2### +

### 12

(33)

*ab*

*b*

*a*

*v*

### =

### 2

2### −

### 6

2### −

### 4

(34) In view of (4),

1 12 6

2 2 − 2 + −

= *a* *b* *ab*

*u*

(35) Substituting (34) and (35) in (2), we get the distinct non- zero integer solutions of (1) is given by

### ( )

### ,

### =

### 4

2### −

### 12

2### +

### 8

### −

### 1

### =

*x*

*a*

*b*

*a*

*b*

*ab*

*x*

### ( )

, =16 −1= *y* *a* *b* *ab*
*y*

### ( )

2 22 2

12 4

,

2 24 12

4 ) , (

*b*
*a*
*b*
*a*
*w*
*w*

*ab*
*b*

*a*
*b*
*a*
*z*
*z*

+ = =

− +

− = =

**PROPERTIES**

1. *x*

### ( )

*a*,

*a*+

*z*(

*a*,

*a*)−16

*t*

_{4}

_{,}

*+3=0*

_{a}64 4.

### 6

### [

*w*

### ( )

### 1

### ,

### 1

### ]

### =

### 6

### ( )

### 4

is a Nasty Number**3.7. PATTERN-7 **

Introduction of the linear transformations
*T*

*X*

*U* = +3 *v*= *X* −*T*
in (3) gives,

2 2 2

### 12

### 4

*X*

### +

*T*

### =

*w*

which can be written as
1 * 12

4*X*2 + *T*2 =*w*2

(36) Write 1 as

16

) 12 2 )( 12 2 (

1= +*i* −*i*

(37) Substituting (37) in (36) and employing the method of factorization,

we get

*ab*
*b*

*a*

*X* = 2 −3 2 −6

*ab*

*b*

*a*

*T*

### =

2### −

### 3

2### +

### 2

*Substituting the values of X,T in (31),we get *

2 2

12
4*a* *b*

*U* = − (38)

*ab*

*v*=−8 (39)
In view of (4),

1 12 4 2 − 2 −

= *a* *b*

*u* (40)
Substituting (39) and (40) in (2), we get the distinct non- zero integer solutions of (1) is
given by,

### ( )

### ,

### =

### 4

2### −

### 12

2### −

### 8

### −

### 1

### =

*x*

*a*

*b*

*a*

*b*

*ab*

*x*

### ( )

### ,

### =

### 4

2### −

### 12

2### +

### 8

### −

### 1

### =

*y*

*a*

*b*

*a*

*b*

*ab*

*y*

### ( )

### ,

### =

### 8

2### −

### 24

2### −

### 2

### =

*z*

*a*

*b*

*a*

*b*

*z*

### ( )

2 2### 12

### 4

### ,

*b*

*a*

*b*

*a*

*w*

*w*

### =

### =

### +

**PROPERTIES **

1. *x*

### (

*a*,

*a*(

*a*+1)

### )

−*y*(

*a*,

*a*(

*a*+1))+16

*CP*

_{n}_{,}

_{6}+16

*t*

_{4}

_{,}

*=0 2.*

_{a}*w*

### (

2*a*+1,

*b*

### )

−28*t*4,

*b*−4≡0

### (

mod2### )

3.

### [

### ( ) ( ) ( )

### ]

### ( )

28 6 4 1 , 1 1 , 1 1 , 1

6*x* +*y* +*z* − = is a Nasty Number

4. *x*

### (

*a*,

*a*+1

### ) (

+*y*

*a*,

*a*+1

### ) (

+*z*

*a*,

*a*+1

### ) (

+*wa*,

*a*+1

### )

−*t*42,

*a*+

*t*74,

*a*+40≡0

### (

mod4### )

**4. Conclusion **

*(x*3*+y*3*)+(x+y)(x+y+1)=zw*2

65

**REFERENCES **

1. *L.E. Dickson, History of Theory of Numbers, Vol 11, Dover publications, New *
York (2005).

2. *L.J.Mordell, Diophantine equations, Academic press, London (1969). *

3. *R.D.Carmicheal, The Theory of Numbers and Diophantine Analysis, Dover *
publications, New York (1959).

4. M.A.Gopalan and V.Pandichelvi, Remarkable solutions of the cubic equations with four unknowns

*x*

3 ### +

*y*

3 ### +

*z*

3 ### =

### 28

### (

*x*

### +

*y*

### +

*z*

### )

*w*

2### ,

*Antartica J Math, 7(4) (2010)*393-401.

5. M.A.Gopalan, S.Premalatha, On the cubic Diophantine equation with four unknowns,

### (

*x*

### −

*y*

### )

### (

*xy*

### −

*w*

2### ) (

### =

### 2

*n*

2 ### +

### 2

*n*

### )

*z*

3*, International Journal of Mathematical*

*Science, 9(2) (2010) 171-175.*

6. M.A.Gopalan, S.Vidhyalakshmi and S.Mallika, Observation on the cubic Diophantine equation with four unknowns,

*xy*

### +

### 2

*z*

2 ### =

*w*

3*, The Global Journal Of*

*Applied Mathematics and Mathematical Science, 2(1) (2012) 69-74.*

7. M.A.Gopalan and K.Geetha, Observation on the cubic Diophantine equation with four unknowns,

*x*

3 ### +

*y*

3 ### +

*xy*

### (

*x*

### +

*y*

### )

### =

*z*

3### +

### 2

### (

*x*

### +

*y*

### )

*w*

2### ,

*International Journal of*

*Pure and Applied Science, 6(1) (2013 ) 25-30.*

8. M.A.Gopalan, S.Vidhyalakshmi and S.Mallika, Observation on the cubic Diophantine equation with four unknowns,

### 2

### (

*x*

3 ### +

*y*

3### )

### =

*z*

3 ### +

*w*

2### (

*x*

### +

*y*

### )

### ,

*JAMP,*4(2) (2012) 103-107.

9. M.A.Gopalan, S.Vidhyalakshmi and G.Sumathi, On the homogeneous cubic equation with four unknowns,

### (

*x*

3### +

*y*

3### )

### =

### 14

*z*

3 ### −

### 3

*w*

2### (

*x*

### +

*y*

### )

*, Discovery, 2(4)*(2012) 7-19.

10. M.A.Gopalan, M.Somanath and V.Sangeetha, Lattice points on the homogeneous cubic equation with four unknowns,

### (

*x*

### +

*y*

### )

### (

*xy*

### +

*w*

2### ) (

### =

*k*

2 ### −

### 1

### )

*z*

3### ,

*k*

### >

### 1

### ,

*Indian*

*Journal of Science, 2(4) (2013) 97-99.*