Silviu Guiasu

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 2, Pages 1—5

DOI: 10.7546/nntdm.2018.24.2.1-5

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## Details

### Authors and affiliations

Silviu Guiasu

*Department of Mathematics and Statistics, York University
4700 Keele Street, Toronto, Ontario M3J 1P3, Canada
*

### Abstract

Due to the fundamental theorem of number theory, the positive integers may be represented by vectors whose components are the unique corresponding powers of the prime numbers. Taking the prime numbers as coordinates, to each positive integer we assign a prime vector, whose components are the powers of the prime factors of this integer. The geometry of this system of prime coordinates of the positive integers is discussed. It is shown that the prime components assigned to the sequence of positive integers change in a strictly deterministic way and the parallel generating system is presented. Gödel’s prime vectors assigned to formal logical formulas are analyzed.

### Keywords

- Prime coordinates
- Prime vectors
- Regular structure of the prime coordinates
- Parallel generating system
- Gödel’s prime vectors

### 2010 Mathematics Subject Classification

- 11A41

### References

- Church, A. (1996) Introduction to Mathematical Logic Princeton University Press, Princeton.
- Gödel, K. (1962) On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Dover Publications, Inc., New York.
- Guiasu, S. (1995) Is there any regularity in the distribution of prime numbers at the beginning of the sequence of positive integers? Mathematics. Magazine, 68 (2), 110–121.
- Guiasu, S. (2009) Probabilistic Models in Operations Research, Nova Science Publishers, New York.
- Nagel, E., & Newman, J. R. (1958) Gödel’s Proof. The New York University Press, New York.

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## Cite this paper

Guiasu, S. (2018). The system of prime coordinates assigned to the positive integers. Notes on Number Theory and Discrete Mathematics, 24(2), 1-5, doi: 10.7546/nntdm.2018.24.2.1-5.