Homogeneous Cubic Equation Lattice Points Integral Solutions Special Polygonal Numbers
= Polygonal number of rank n
with m sides. = Three-dimensional figurate
number of rank n with m sides. = Star number. = Rhombic Dodecahedral number of rank n. The Homogeneous cubic equation with four unknowns represented by the equation is analyzed for its patterns of non zero distinct integral solutions. Here we exhibit four different patterns. In each pattern we can find some interesting relations between the solutions and special numbers like Polygonal number, Three-Dimensional Figurate number, Star number, Rhombic Dodecahedral number etc.
## 1. INTRODUCTIONNumber theory, called the Queen of Mathematics, is a broad and diverse part of Mathematics that developed from the study of the integers. Diophantine equation is one of the oldest branches of Mathematics. Diophantine problems dominated most of the unsolved mathematical problems. The cubic equation offers an unlimited field of research because of their variety. This paper concerns with an interesting equation , representing a homogeneous cubic equation with four unknowns for finding its infinitely many solutions and some interesting relations between the solutions and special numbers like Polygonal number, Rhombic Dodecahedral number, Star number, Three Dimensional Figurate number. ## 2. METHOD OF ANALYSISConsider the ## 2.1. PATTERN 1Introduction of the transformation in leads to Assume and write (5) Using and in and using method of factorization, define
Equating real and imaginary parts on both sides of , we get
Substituting and in , we obtain the solutions of as
Some properties for the above solution are listed below:
is a Nasty Number
## 2.2. PATTERN 2Introduction of the transformation in leads to
Assume and write (10) Using and in and using method of factorization, define (11) Equating real and imaginary parts on both sides of , we get (12)
Since our aim is to find integer values for the solution, put and inand , we get (14) (15) Substituting and in , we obtain the solutions of as
Some properties for the above solution are listed below:
. is a Nasty Number.
## 2.3. PATTERN 3Introduction of the transformation in leads to
Assume and write (19) Usingand in and using method of factorization, define
Equating real and imaginary parts on both sides of , we get (21)
Since our aim is to find integer values for the solutions, put and inand , we get (23) (24) Substituting and in , we obtain the solutions of as
Some properties for the above solution are listed below:
## 2.4. PATTERN 4Introduction of the transformation in leads to
Put in ,
we get (28) Assume and write (30) Using and in and using method of factorization, define
Equating real and imaginary parts on both sides of , we get (32) Since our aim is to find integer values for the solutions, put and inand , we get (34) (35)
Substituting and in , we obtain the solutions of as
Some properties for the above solution are listed below:
, a Nasty Number , a Perfect Square ## 3. CONCLUSIONDiophantine Equation are rich in variety. To conclude, one may search for several other patterns of solutions and their properties. ## SOURCES OF FUNDINGThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. ## CONFLICT OF INTERESTThe author have declared that no competing interests exist. ## ACKNOWLEDGMENTNone. ## REFERENCES
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