Introduction to Non-Diophantine Number Theory

Mark Burgin

Abstract


In the 19th century, non-Euclidean geometries were discovered and studied. In the 20th century, non-Diophantine arithmetics were discovered and studied. Construction of non-Diophantine arithmetics is based on more general mathematical structures, which are called abstract prearithmetics, as well as on the projectivity relation between abstract prearithmetics. In a similar way, as set theory gives a foundation for mathematics, the theory of abstract prearithmetics provides foundations for the theory of the Diophantine and non-Diophantine arithmetics. In this paper, we use abstract prearithmetics for developing fundamentals of non-Diophantine number theory, which can be also called non-Diophantine higher arithmetic as the conventional number theory is called higher arithmetic. In particular, we prove the Fundamental Theorem of Arithmetic for a wide range of abstract prearithmetics.


Keywords


number, arithmetic, number theory, divisibility, prime number, factorization, addition, difference, multiplication

Full Text:

PDF

References


Aristotle (1984). The Complete Works of Aristotle. Princeton University Press, Princeton.

Baranovich, T.M. and M. Burgin (1975). Linear ω-algebras. Russian Mathematical Surveys 30(4), 61–106.

Beechler, D. (2013). How to create 1 + 1= 3 marketing campaigns. http://www.marketingcloud.com/blog/ how-to-create-1-1-3-marketing-campaigns.

Broadbent, T.A.A. (1971). The higher arithmetic. Nature Physical Science 229(6), 187–188.

Brodsky, Anne E, Kathleen Rogers Senuta, Catharine L A Weiss, Christine M Marx, Colleen Loomis, S Sonia Arteaga, Heidi Moore, Rona Benhorin and Alisha Castagnera-Fletcher (2004). When one plus one equals three: the role of relationships and context in community research. American journal of community psychology 33(3-4), 229–241.

Burgin, M. (1977). Nonclassical models of the natural numbers. Uspekhi Mat. Nauk 32(6(198)), 209–210.

Burgin, M. (1997). Non-Diophantine Arithmetics or what number is 2+2 ?. Ukrainian Academy of Information Sciences, Kiev. (in Russian, English summary).

Burgin, M. (2001). Diophantine and non-Diophantine arithmetics: Operations with numbers in science and everyday life. LANL, Preprint Mathematics GM/0108149, 27 p. (electronic edition: http://arXiv.org).

Burgin, M. (2007). Elements of non-Diophantine arithmetics. In: 6th Annual International Conference on Statistics, Mathematics and Related Fields, 2007 Conference Proceedings, Honolulu, Hawaii. pp. 190–203.

Burgin, M. (2010). Introduction to projective arithmetics. Preprint in Mathematics, math.GM/1010.3287, 21 p. (elec- tronic edition: http://arXiv.org).

Burgin, M. (2012). Hypernumbers and Extrafunctions: Extending the Classical Calculus. Springer, New York.

Burgin, M. and G. Meissner (2017). 1 + 1 = 3: Synergy arithmetic in economics. Applied Mathematics 08, 133–144.

Bussmann, Johannes (2013). One plus one equals three (or more ): combining the assessment of movement behavior and subjective states in everyday life. Frontiers in Psychology 4, 216.

Cleveland, A. (2008). One plus one doesn’t always equal two. Circadian Math.

Czachor, Marek (2016). Relativity of arithmetic as a fundamental symmetry of physics. Quantum Studies: Mathemat- ics and Foundations 3(2), 123–133.

Czachor, Marek (2017a). If gravity is geometry, is dark energy just arithmetic?. International Journal of Theoretical Physics 56(4), 1364–1381.

Czachor, Marek (2017b). Information processing and Fechner's problem as a choice of arithmetic. Information Studies and the Quest for Transdisciplinarity: Unity through Diversity, World Scientific, New York/London/Singapore pp. 363–372.

Czachor, Marek and Andrzej Posiewnik (2016). Wavepacket of the universe and its spreading. International Journal of Theoretical Physics 55(4), 2001–2019.

Davenport, H. (1999). The Higher Arithmetic: An Introduction to the Theory of Numbers. Cambridge University Press.

Davis, P. J. (1972). Fidelity in mathematical discourse: Is one and one really two?. The American Mathematical Monthly 79(3), 252–263.

Davis, P.J. and R. Hersh (1998). The Mathematical Experience. Mariner books. Houghton Mifflin.

Derboven, J. (2011). One plus one equals three: Eye-tracking and semiotics as complementary methods in HCI. CCID2: The Second International Symposium on Culture, Creativity, and Interaction Design, Newcastle, UK.

Enge, E. (2017). Seo and social: 1 + 1 = 3, SearchEngineLand. https://searchengineland.com/ seo-social-1-1-3-271978.

Fraenkel, A.A., Y. Bar-Hillel and A. Levy (1973). Foundations of Set Theory. Studies in Logic and the Foundations of Mathematics. Elsevier Science.

Frame, A. and P. Meredith (2008). One plus one equals three: Legal hybridity in Aotearoa/New Zealand. Hybrid Identities pp. 313–332.

Fuchs, L. (1963). Partially Ordered Algebraic Systems. Dover Books on Mathematics. Pergamon Press, Oxford/London/New York/Paris.

Gardner, M. (2005). Review of science in the looking glass: What do scientists really know? By

E. Brian Davies (Oxford University Press, 2003). Notices of the American Mathematical Society. (http://www.ams.org/notices/200511/rev-gardner.pdf).

Glyn, A. (2017). One plus one equals three the power of data combinations, Luciad. https://searchengineland. com/seo-social-1-1-3-271978.

Gottlieb, A. (2013). ’1 + 1 = 3’: The synergy between the new key technologies, Next-generation Enterprise WANs. Network World. http://www.networkworld.com/article/2224950/cisco-subnet/

-1---1---3---the-synergy-between-the-new-key-technologies.html.

Grant, M. and C. Johnston (2013). 1 + 1 = 3: CMO & CIO collaboration best practices that drive growth. Canadian Marketing Association, Don Mills, Canada.

Hayes, B. (2009). The higher arithmetic. American Scientist 97(5), 364–367.

Helmholtz, H. von (1887). Za¨hlen und Messen In: Philosophische Aufsatze, pp. 17-52, (Translated by C.L. Bryan, ”Counting and Measuring”, Van Nostrand, 1930). Verlag: Leipzig: Fues‘s Verlag (R. Reisland).

Hilbert, David (1899). Grundlagen der Geometrie, Festschrift zur Feier der Enthllung des Gauss-Weber Denkmals.

Gttingen, Teubner, Leipzig.

Klees, E. (2006). One Plus One Equals Three - pairing Man/Woman Strengths: Role Models of Teamwork (The Role Models of Human Values Series, Vol. 1). Cameo Press, New York.

Kline, M. (1982). Mathematics: The Loss of Certainty. A Galaxy book. Oxford University Press.

Kline, M. (1985). Mathematics for the Nonmathematician. Addison-Wesley series in introductory mathematics.

Dover.

Kroiss, Matthias, Utz Fischer and Jrg Schultz (2009). When one plus one equals three: Biochemistry and bioinformatics combine to answer complex questions. Fly 3, 212–214.

Kurosh, A.G. (1963). Lectures in general algebra:. number vol. 70 In: International series of monographs in pure and applied mathematics. Chelsea P. C., New York.

Landau, E., P.T. Bateman and E.E. Kohlbecker (1999). Elementary Number Theory. AMS Chelsea Publishing Series.

American Mathematical Society.

Lang, M. (2014). One plus one equals three. Optik & Photonik 9(4), 53–56.

Lawrence, C. (2011). Making 1 + 1 = 3: Improving sedation through drug synergy. Gastrointestinal Endoscopy.

Lea, R. (2016). Why one plus one equals three in big analytics, Forbes. https://www.forbes.com/sites/ teradata/2016/05/27/why-one-plus-one-equals-three-in-big-analytics/#1aa2070056d8.

Mane, R. (1952). Evolution of mutuality: one plus one equals three; formula characterizing mutuality. La Revue Du Praticien 2(5), 302–304.

Marie, K.L. (2007). One plus one equals three: Joint-use libraries in urban areas - the ultimate form of library coop- eration. Library Administration and Management 21(1), 23–28.

Marks, M.L. and P.H. Mirvis (2010). Joining Forces: Making One Plus One Equal Three in Mergers, Acquisitions, and Alliances. Wiley.

Nieuwmeijer, C. (2013). 1 + 1 = 3: The positive effects of the synergy between musician and classroom teacher on young childrens free musical play. Dissertation, Roehampton University, London.

Phillips, J. (2016). When one plus one equals three, wellness universe. http://blog.thewellnessuniverse.com/ when-one-plus-one-equals-three.

Robinson, A. (1966). Non-Standard Analysis, Studies of Logic and Foundations of Mathematics. North-Holland, New York.

Trabacca, A., G. Moro, L. Gennaro and L. Russo (2012). When one plus one equals three: The ice perspective of health and disability in the third millennium. European Journal of Physical and Rehabilitation Medicine 48(4), 709–710.

Trott, D. (2015). One Plus One Equals Three: A Masterclass in Creative Thinking. Macmillan Publishing Company, New York.

Veronese, G. (1889). Il continuo rettilineo e lassioma V di Archimede. Memorie della Reale Accademia dei Lincei, Atti della Classe di scienze naturali, fisiche e matematiche 6(4), 603–624.


Refbacks

  • There are currently no refbacks.