# Polar Integers and Polar Integer Optimization

### Abstract

This is a pioneering work, introducing a special class of complex numbers, wherein their absolute values and arguments given in a Polar coordinate system are integers, which when considered within the complex plane, constitute Unicentered Radial Lattice and similarly for quaternions and Euclidean R2 and R3 Spaces. The corresponding Optimization Problems are introduced as well.### References

M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Annals of Mathematics, vol. 160, pp. 781–793, 2004.

S. Bernstein, U. Kahler, I. Sabadini and F. Sommen, Hypercomplex Analysis: New Perspectives and Applications, Birkhauser, 2014.

L. M. B. C. Campos, Complex Analysis with Applications to Flows and Fields, CRC Press, 2011.

J. Coolidge, The Origin of Polar Coordinates, American Mathematical Monthly, Mathematical Association of America, vol. 59, no. 2, pp. 78–85, 2018.

T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction To Algorithms, The MIT Press, Cambridge, 2009.

C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization, Springer, New York, 2009.

I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics, Advances in Mathematics, vol. 218, pp. 1806–1877, 2008.

B. Green and T. Tao, Linear equations in primes, Annals of Mathematics, vol.171, pp. 1753–1850, 2010.

R. Hemmecke, M. Koppe, J. Lee and R. Weismantel, ¨ Nonlinear Integer Programming, in 50 Years of Integer Programming 1958–2008: The Early Years and State-of-the-Art Surveys (eds. M. Junger, T. Liebling, D. Naddef, W. Pulleyblank, W. Reinelt, G. Rinaldi and Wolsey), Springer-Verlag, Berlin, pp. 561–618, 2010.

G. James, Modern Engineering Mathematics, Trans-Atlantic Pubns Inc., 2015.

I. Kleiner, From Numbers to Rings: The Early History of Ring Theory, Elem. Math, Birkhauser, Basel, vol. 53, pp. 18–35, 1998.

E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc., 2011.

P. Moon, D. E Spencer, Circular-Cylinder Coordinates (r, ψ, z). Field Theory Hand-book, Including Coordinate Systems, Differential Equations, and Their Solutions, (corrected 2nd ed.). New York City: Springer-Verlag., pp. 12–17, 1989.

P. Moon, D. E Spencer, Spherical Coordinates (r, θ, ψ). Field Theory Hand-book, Including Coordinate Systems, Differential Equations, and Their Solutions, (corrected 2nd ed., 3rd print ed.). New York City: Springer-Verlag., pp. 24–27, 1988.

V. Neale, Closing the Gap: The Quest to Understand Prime Numbers, Oxford University Press, 2017.

V. Scheidemann, Introduction to complex analysis in several variables, Birkhauser, 2005.

W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006.

Y. Shipilevsky, Complex and quaternionic optimization, Numerical Algebra, Control and Optimization, . vol. 10, no. 3, pp. 249–255, 2020.

Y. Shipilevsky, Prime Optimization, Statistics, Optimization, & Information Computing, vol. 9, no. 2, pp. 453–458, 2021.

V. Scheidemann, Introduction to complex analysis in several variables, Birkh¨auser, 2005.

W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006.

Y. Shipilevsky, Complex and quaternionic optimization, Numerical Algebra, Control and Optimization, . vol. 10, no. 3, pp. 249–255, 2020.

Y. Shipilevsky, Prime Optimization, Statistics, Optimization, & Information Computing, vol. 9, no. 2, pp. 453–458, 2021.

*Statistics, Optimization & Information Computing*,

*10*(3), 925-934. https://doi.org/10.19139/soic-2310-5070-1327

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