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This paper takes an interesting approach to conceptualize
some power sum inequalities and uses them to develop limits on
possible solutions to some Diophantine equations. In this work, we
introduce how to apply center of mass of a k-mass-system to discuss a
class of Diophantine equations (with fixed positive coefficients) and
a class of equations related to Fermatâs Last Theorem. By a constructive
method, we find a lower bound for all positive integers that
are not the solution for these type of equations. Also, we find an
upper bound for any possible integral solution for these type of equations.
We write an alternative expression of Fermatâs Last Theorem
for positive integers in terms of the product of the centers of masses
of the systems of two fixed points (positive integers) with different
masses. Finally, by assuming the validity of Bealâs conjecture, we
find an upper bound for any common divisor of x, y, and z in the
expression ax^m +by^n = z^r
in terms of a, b, m(or n), r, and the center
of mass of the k-mass-system of x and y.
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