Introduction to Non-Diophantine Number Theory

Mark Burgin


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.


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

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