An Innovative Approach to the Finite Sequences of Prime Numbers

Main Article Content

Daniele Lattanzi


An innovative approach that treats prime numbers as raw experimental data making use of experimental/computational mathematics and the approximation methods is presented in order to get advanced and more exact formulations of the canonical form  =    being the prime value and  its counter. The use of many different functions - such as the inverse of the modified chi-square function  with its three parameters ,  and , the function  with the ad-hoc  values being  , the function , the function , the harmonic series  and its approximation by Euler and so on - as fit functions of finite sets i.e. sequences of prime numbers leads to induction algorithms and to new relationships of the kind  though within the approximations of the calculations with all the estimations better than that of the standard formulation . In such a manner, refined formulations with higher precisions are got showing that there are many ways to treat the finite sequences of prime numbers. Comparisons among the various methods are made in order to find the best formulation of a new and more refined relationship in a closed form that can be valid to find the most approximate value of a prime starting from its counter in the finite case.

Prime number sequences, data fits, modified chi-square function, experimental mathematics, computational mathematics.

Article Details

How to Cite
Lattanzi, D. (2020). An Innovative Approach to the Finite Sequences of Prime Numbers. Journal of Advances in Mathematics and Computer Science, 35(9), 34-56.
Original Research Article


Smith DE. A source book in mathematics, Dover Publications, Inc. New York, unabridged republication of the 1rst edition, originally published in 1929 by McGraw-Hill Book Co., Inc; 1959.

Karl-Heinz Kuhl K-H, Prime Numbers – things long-known and things new-found, Parkstein. Eckhard Bodner, Pressath, Germany; 2019.

Cipolla M. La determinazione asintotica del nmo numero primo. Matematiche Napoli. 1902;3:132-166. Italian.

Babusci D, Dattoli G, Del Franco M. Lectures on mathematical methods for physics. RT/2010/58/ENEA, ENEA-Roma-I.

Fine B, Rosenberger G. Number Theory - An Introduction via the Distribution of Primes Birkäuser Boston, USA; 2007.

Du Sautoy M. L’enigma dei numeri primi RCS Libri S.p.A. Milano. ISBN 978-8817-05022-7. Italian; 2004.

Derbyshire J. L’ossessione dei numeri primi – Bernhard Riemann e il principale problema irrisolto della matematica. Bollati Boringhieri Editore s.r.l. Torino. Italian; 2006.

Languasco A, Zaccagnini A. Alcune proprietà dei numeri primi, I e II, Sito web Bocconi-Pristem; 2005. Italian.

Zaccagnini A. Introduzione alla teoria analitica dei numeri. Italian. Università degli Studi di Parma, Facoltà di Scienze Matematiche, Fisiche e Naturali, Corso di Laurea in Matematica; 2005.

Green B. Long arithmetic progressions of primes. Clay Mathematics Proceedings. 2007;7:149–167.

Green B, Tao T. The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics. 2008;167(2):481-547.
Available: http://www.arXiv:math.NT/0404188

Goldoni L. PhD Thesis, Prime Numbers and Polynomials, Università degli Studi di Trento, Facoltà di Scienze Matematiche, Fisiche e Naturali, Dottorato di ricerca in Matematica, XXIII ciclo, Academic; 2009 – 2010.

Wells D. Prime numbers - the most mysterious figures in math, john wiley and sons, Inc; 2005.

Porras Ferreira JW. The pattern of prime numbers. Applied Mathematics. 2017;8:180-192.

Tapia-Moore E. Y Tapia-Yañez J. The occurrence of prime numbers revisited. 2016;4(1):2016. GECONTEC: Revista Internacional de Gestión del Conocimiento y la Tecnología. ISSN 2255-5684

Jensen JH. Subtle Relations: Prime Numbers, Complex Functions, Energy Levels and Riemann, Available:

Bershadskii A. Hidden periodicity and chaos in the sequence of prime numbers. Hindawi Publishing Corporation Advances in Mathematical Physics; 2011. Article ID 519178.

Sierra G. in collaboration with Latorre JI, Madrid-Barcelona-Singapore, Primes go Quantum: there is entanglement in the primes, workshop: Entanglement Entropy in Quantum Many-Body Systems, King's College and City University, London. 2014;arXiv:1302.6245 and arXiv:1403.4765

Knill O. On Particles and Primes. 2016;arXiv:1608.07175v1 [physics.gen-ph].

Marcolli M. Caltech, geometry and physics of numbers, Revolution Books, Berkeley; 2013.

Torquato S, Zhang G, de Courcy-Ireland M, Stat J. Mech. Uncovering multiscale order in the prime numbers via scattering. 2018;093401.
Available: arXiv:physics/0005067v2 [physics.gen-ph],10 Aug 2000

Pitkänen M. Physics as generalized number theory: Infinite Primes; 2010.

Selvam AM. Quantum-like chaos in prime number distribution and in turbulent fluid flows, Indian Institute of Tropical Meteorology Pune 411 008, India; 2000.
Available: arXiv:physics/0005067v2 [physics.gen-ph]

Kelly DT. From prime numbers to nuclear physics and beyond By Kelly Devine Thomas Published in The Institute Letter, IAS, Spring; 2013.

Cipra B. A prime case of chaos. A Prime Case of Chaos" by Barry Cipra, AMS, American Mathematical Society; 2003.

Brewer G. Prime symmetry and particle physics. Matador, 9 Priory Business Park, Wistow Road, Kibworth Beauchamp, Leicestershire, U.K; 2017.

Bach E. Intractable problems in number theory. In: Goldwasser S. (eds) Advances in Cryptology — CRYPTO’ 88. Lecture Notes in Computer Science. 1990;403. Springer, New York, NY.

Sahli A. The ultimate solution approach to intractable problems, Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010). Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia; 2010.

Kotnik T. The prime-counting function and its analytic approximations - π(x) and its approximations. Adv Comput Math. 2008;29:55–70.
DOI 10.1007/s10444-007-9039-2 © Springer Science + Business Media B.V. 2007

Liberatore P. Compilation of intractable problems and its application to artificial intelligence, Università degli Studi di Roma “La Sapienza. Dottorato di ricerca in Ingegneria Informatica, X Ciclo, Italian; 1998.

Lattanzi D. Distribution of prime numbers by the modified chi-square function. Notes on Number Theory and Discrete Mathematics. 2015;21(1):18-30.

Lattanzi D. Scale laws of prime number frequencies by the modified chi-square function. Journal of Advances in Mathematics and Computer Science, former British Journal of Mathematics and Computer Science. 2016;13(6):1-21. Article no.BJMCS.23200.
DOI: 10.9734/BJMCS/2016/23200

Lattanzi D. Computational model of prime numbers by the modified chi-square function. Journal of Advances in Mathematics and Computer Science, former British Journal of Mathematics and Computer Science. 2017;20(5):1-19. Article no.BJMCS.31589, ISSN:2231-0851
DOI: 10.9734/BJMCS/2016/31589

Stallinga P. Scalable Functions Used for Empirical Forecasting, former British Journal of Mathematics and Computer Science, now Journal of Advances in Mathematics and Computer Science, former British Journal of Mathematics and Computer Science. 2016;18(2). Article no.BJMCS.28107

Ghosh A. Mechanics over Micro and Nano Scales, Chapter 2, Scaling Laws, Editor Chakraborthy S. 2011;IX:269. ISBN 978-1-4419-96008

Lattanzi D. An Elementary Proof of Riemann’s Hypothesis by the Modified Chi-square Function. Journal of Advances in Mathematics and Computer Science, former British Journal of Mathematics and Computer Science. 2016;15(05):1-14. Article no. BJMCS.25419.
DOI: 10.9734/BJMCS/2016/25419

Bailey DH, Borwein JM. Exploratory experimentation and computation August 14, 2010 - LBNL Paper LBNL-3313E Lawrence Berkeley National Laboratory; Notices of the AMS. 2011;58(10):1410-1419.

Andeberhan T, Medina LA, Moll VH. Editors, contemporary mathematics - 517 – Gems in Experimental Mathematics. AMS Special Session, Experimental Mathematics. Washington D.C. American Mathematical Society; 2009.

Thomas Oliveira e Silva, Dep. De Electronica, Telecomuniçaões e Informatica, Universidade de Aveiro, Portugal Goldbach Conjecture verification; Available:

Sørensen H.K., Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm Institut for Videnskabsstudier, Aarhus Universitet, 8000 Ärhus C, Denmark.

Borwein J.M., Exploratory Experimentation: Digitally-Assisted Discovery and Proof, May 18, 2009, University of Newcastle, Australia.

Saeli D. and Spano M, La cometa di Goldbach e le altre. Lecture Notes of Seminario Interdisciplinare di Matematica. 2011;10:45–57. Italian.

Sørensen HK. Experimental mathematics in the 1990s: A second loss of certainty?, Disciplines and Styles in Pure Mathematics, 1800-2000, Oberwolfach Report 12/2010. 2010;601-604.

Crandall R, Pomerance C. Prime numbers A computational perspective. Springer Science+Business Media, Inc; 2005.

Brown JR. Philosophy of Mathematics - A Contemporary Introduction to the World of Proofs and Pictures, Second Edition, Routledge, Francis & Taylor Group, New York and London; 2008.

Epstein D, Levy S. Experimentation and proof in mathematics. Notices of the AMS. 1995;42(6).

Porter MA, Zabusky NJ, Hu B, Campbell DK. Fermi, Pasta, Ulam and the Birth of Experimental Mathematics, A reprint from American Scientist - the magazine of Sigma Xi, The Scientific Research Society, American Scientist. Sigma Xi, The Scientific Research Society. 2009;97.

Baker A. Stanford encyclopedia of philosophy. Non-Deductive Methods in Mathematics; 2009.






Sloane NJA. The on-line encyclopaedia of integer sequences© ((OEIS©)

Young HD. Carnegie Institute of Technology - Statistical Treatment of Experimental Data, McGraw-Hill Book Company, Inc; 1962.

Morice E. Dizionario di statistica by ISEDI. Milano. Italian; 1971.

Mathews J, Walker RL. CalTech - mathematical methods of physics W.A. Benjamin Inc. New York N.Y; 1964.

Walck C. Hand-book on Statistical Distributions for Experimentalists. Internal Report SUF–PFY/96–01 Stockholm, 10 September 2007.

Kowalski E. Arithmetic Randonnée - An introduction to probabilistic number theory; 2020.
Available: [email protected]

Yeseul K. Byung Mook Weon, Randomness at large numbers: Experimental proof in coin toss and prime number. AIP Advances. 2019;9:125318.
Available DOI: 10.1063/1.5133773

Luque B, Lacasa l. The first-digit frequencies of prime numbers and Riemann zeta zeros, Proc. R. Soc. A, published online 22 April 2009.
DOI: 10.1098/rspa.2009.0126 Available:

Gronau QF, Wagenmakers E. Bayesian evidence accumulation in experimental mathematics: A Case Study of Four Irrational Numbers, Experimental Mathematics. 2020;27:3:277-286.
DOI: 10.1080/10586458.2016.1256006

Mariconda C, Tonolo A. Discrete calculus – methods for counting, UNITEXT 103, © Springer International Publishing Switzerland; 2016. ISBN 978-3-319-03037