# On the Universal Encoding Optimality of Primes

## Abstract

**:**

## 1. Introduction

## 2. ILP Formulation–Additive Optimality of Prime Factors

^{≥2}, and, in that sense, P is irreducible in a quantum sense; i.e., no subset of ℤ with smaller cardinality or member values exists having the same universally optimal factorial encoding capability in ℤ.

**Definition**

**1.**

^{≥2}, the Prime Counting Function π(s) is equal to the number of primes less than or equal to s.

**Notation.**

^{≥2}are used in this paper as representations of the set of real numbers, integers and integers greater or equal to 2, respectively. The nth prime number, in ascending order, is denoted as ${p}_{n}$; P

_{n}is the set of the first n primes, P of all primes, $\widehat{P}=P\cup \{1\}$, ${P}_{o}$ is the set of odd primes, and ${\tilde{P}}_{o}={P}_{o}\cup \{1\}$. The integer part of x ∈ ℝ, i.e., the largest integer less than or equal to x, is denoted by ⌊x⌋ and the discrete delta function, taking the values of 1 for x = 0 and 0 for x ≠ 0, is denoted by δ(x). The natural logarithm of x is log(x).

**Example**

**1.**

^{≥3}with π(s) = n for some n ∈ ℤ

^{≥1}; then s may be expressed by its prime factorization

^{≥0}for i ∈ [1, n]. By taking the natural logarithm on both sides of (1)

**Example**

**2.**

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Example**

**3.**

**Example**

**4.**

**Remark**

**3.**

**Remark**

**4.**

_{n}, given by (7), had any prime factors ≤ n, it would follow that ${2}^{m}$ would have at least one of those as prime factor; false, since the only prime factor of ${2}^{m}$ is 2.

**Example**

**4.**

**Remark**

**5.**

**Remark**

**6.**

^{≥0}for i∈ [1, n] are the factor exponents and since, by definition, all${p}_{i}$≥ 2

**Remark**

**7.**

## 3. Summative Optimality of Primes

**Example**

**5.**

_{o}× P

_{o}pairs, marked with *.

_{o}× P

_{o}pairs more efficient. We discuss this in Appendix B. Note that a consequence of the prime density decreasing with s is that—even for large values of s—primes are not evenly distributed in [1, s/2] and [s/2, s]; this is discussed in Appendix C.

## 4. Quantum Encoding Optimality

**Corollary**

**1.**

**Proof.**

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Two Closed-Form Expressions for the Prime Counting Function and an Algebraic Formulation of the Goldbach Conjecture

**Proposition**

**A1.**

^{>3}. Then s is prime if and only if ${T}_{s}$= 0, where

**Proof.**

_{s}is nonzero if and only if s is divisible by some integer in the interval [2, $s/2$], i.e., s is not prime. Therefore, T

_{s}= 0 can only occur if s is prime. □

_{s}≠ 0, composite s) while prime valued inputs are blocked (T

_{s}= 0, prime s).

**Proposition**

**A2.**

^{>3}. The prime counting functionπ(s) is given by

**Proof.**

_{j})⌋ takes the value 1 if j is prime (Τ

_{j}= 0), or the value of 0 if j is not prime (Τ

_{j}= 1), and the proposition is proved. □

**Proposition**

**A3.**

^{>3}

**Proof.**

## Appendix B. Efficient Searches for Prime-Prime (P-P) Type Pairs within a Partition

s/2 | s mod 6 = 0 | s mod 6 = 2 | s mod 6 = 4 |
---|---|---|---|

even | f = 3λ ± 1, λ ≥ 0 even | f = 3λ, λ ≥ 1 odd f = s/2 − 3 | f = 3λ, λ ≥ 1 odd f = s/2 − 3 |

odd | f = 3λ ± 1, λ ≥ 1 odd | f = 3λ, λ ≥ 0 even | f = 3λ, λ ≥ 0 even f = s/2 − 3 |

## Appendix C. On the Relative Density of Primes between [1, x] and [x, 2x]

^{16}) = 279,238,341,033,925 and π(2·10

^{16}) = 547,863,431,950,008, documented in [16], so that π(10

^{16})/π(2·10

^{16}) ≈ 50.9686%, or equivalently, R(10

^{16}) = 49.0314%, which implies a difference of approximately 10.6 M in the number of primes found between [1, 10

^{16}] and [10

^{16}, 2·10

^{16}], or about 1.94% of π(2·10

^{16}).

## References

- Aristotle. Metaphysics; Book 1 sec. 985b–986a, Perseus Digital Library; Tufts University, Massachusetts, USA: Translated in English by Hugh Tredennick: “they (ref. to the Pythagoreans) assumed the elements of numbers to be the elements of everything, and the whole universe to be a proportion (‘Harmony’) or number” and in the Original Greek Text: “τὰ τῶν ἀριθμῶν στοιχεῖα τῶν ὄντων στοιχεῖα πάντων ὑπέλαβον εἶναι, καὶ τὸν ὅλον οὐρανὸν ἁρμονίαν εἶναι καὶ ἀριθμόν.”. Available online: http://data.perseus.org/citations/urn:cts:greekLit:tlg0086.tlg025.perseus-eng1:1.985b (accessed on 29 November 2021).
- Laertius, D. Lives of Eminent Philosophers; Henderson, J., Ed.; Hicks, R.D., Translator; LOEB Classical Library Series; Harvard University Press: Cambridge, MA, USA, 2005; Volume II, pp. 340–347. [Google Scholar]
- Guthrie, K. The Pythagorean Sourcebook and Library; Alexandria Books; Phanes Press: Grand Rapids, MI, USA, 1988; pp. 299–305, 321–331. [Google Scholar]
- Laks, A.; Most, G. Early Greek Philosophy; LOEB Classical Library Series; Fragment D50 (B54) “Invisible Harmony, Stronger Than a Visible One”; and in the Original Greek Text: “ἁρμονίη ἀφανὴς φανερῆς κρείττων.”; Harvard University Press: Cambridge MA, USA, 2016; Volume III, p. 162. [Google Scholar]
- Euclid. Elements, Propositions 7.32 (p. 219) and 9.20 (p. 271) from the Greek Text of J. L. Heiberg (1883–1885) based on Euclidis Elementa, Edited and Translated in English by Richard Fitzpatrick. Available online: http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf (accessed on 29 November 2021).
- Hawking, S. God Created the Integers; Propositions 7.32 (p. 92) and 9.20 (p. 101); Running Press: Philadelphia, PA, USA, 2005. [Google Scholar]
- Derbyshire, J. Prime Obsession; Plume: Washington, DC, USA, 2004; pp. 99–101. [Google Scholar]
- Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers, 6th ed.; Oxford University Press: New York, NY, USA, 2008; p. 23. [Google Scholar]
- Rivest, R.L.; Shamir, A.; Adleman, L. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Commun. ACM
**1978**, 21, 120–126. Available online: http://people.csail.mit.edu/rivest/Rsapaper.pdf (accessed on 29 November 2021). [CrossRef] - Chaitin, G. Meta Math! The Quest for Omega; Vintage Books: New York, NY, USA, 2006; pp. 13–22. [Google Scholar]
- Wikipedia. Goldbach’s Conjecture. Available online: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture (accessed on 29 November 2021).
- Caldwell, K. The Prime Pages: Prime Conjectures and Open Questions. Available online: https://primes.utm.edu/notes/conjectures/ (accessed on 29 November 2021).
- Oliveira e Silva, T.; Herzog, S.; Pardi, S. Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10
^{18}. Math. Comput.**2014**, 83, 2033–2060. Available online: https://www.ams.org/journals/mcom/2014-83-288/S0025-5718-2013-02787-1/S0025-5718-2013-02787-1.pdf (accessed on 29 November 2021). [CrossRef][Green Version] - Caldwell, K. The Prime Pages: Goldbach’s Conjecture. Available online: https://primes.utm.edu/glossary/page.php?sort=GoldbachConjecture (accessed on 29 November 2021).
- Weisstein, E.W. Prime Counting Function. MathWorld—A Wolfram Web Resource. Available online: https://mathworld.wolfram.com/PrimeCountingFunction.html (accessed on 29 November 2021).
- Gourdon, X.; Sebah, P. Numbers, Constants and Computation. Available online: http://numbers.computation.free.fr/Constants/Primes/pixtable.html (accessed on 29 November 2021).

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Papadakis, I.N.M.
On the Universal Encoding Optimality of Primes. *Mathematics* **2021**, *9*, 3155.
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On the Universal Encoding Optimality of Primes. *Mathematics*. 2021; 9(24):3155.
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Papadakis, Ioannis N. M.
2021. "On the Universal Encoding Optimality of Primes" *Mathematics* 9, no. 24: 3155.
https://doi.org/10.3390/math9243155