Article Type: Research Article Article Citation: Manju Somanath,
Radhika Das, and Bindu V.A. (2020). LATTICE POINTS ON THE HOMOGENEOUS CUBIC
EQUATION WITH FOUR UNKNOWNS Received Date: 30 July 2020 Accepted Date: 26 August 2020 Keywords: Homogeneous Cubic Equation Lattice Points Integral Solutions Special Polygonal Numbers Notation
The Homogeneous cubic equation with four unknowns
represented by the equation
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
, 2. METHOD OF ANALYSISConsider the Homogeneous
cubic equation, 2.1. PATTERN 1Introduction of the transformation Assume Using
Equating real and imaginary parts on both sides of
Substituting
Some properties for the above solution are listed below:
2.2. PATTERN 2Introduction of the transformation
Assume Using
Equating real and imaginary parts on both sides of
Since our aim is to find integer values for the solution,
put
Substituting
Some properties for the above solution are listed below:
2.3. PATTERN 3Introduction of the transformation
Assume Using
Equating real and imaginary parts on both sides of
Since our aim is to find integer values for the solutions,
put
Substituting
Some properties for the above solution are listed below: 2.4. PATTERN 4Introduction of the transformation
Put Assume Using
Equating real and imaginary parts on both sides of
Since our aim is to find integer values for the solutions,
put
Substituting
Some properties for the above solution are listed below:
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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