MODELS OF THE FUNDAMENTAL THEOREM OF ARITHMETIC

This paper announces various extensions of the author's new formulation of the "Fundamental Theorem of Arithmetic" (abbreviated FTA) 1-I so as to classify all multiplicative models of FTA and to show new relations between additive number theory and multiplicative number theory. Further, certain connections are given between the theory to be presented, and algebra (especially ideal theory a la Kummer-Kronecker-Hilbert) and analysis (as concern limiting distributions for squarefree numbers). In general, proofs are not given here. 1. The basic method of the author is reflexive induction, i.e., applying a rule to a portion of the entity obtained from a prior application of that same rule. If a reflexive induction terminates in a finite number of steps, it is called a finite reflexive induction, otherwise an infinite reflexive induction. For example, the Euclidean Algorithm is obtained from the Division Algorithm by finite reflexive induction upon the Division Algorithm; in full strength, Fermat's method of infinite descent4 makes use of an infinite reflexive induction to obtain the standard kind of contradiction for a reductio ad absurdum proof. The new formulation of FTA uses finite reflexive induction in that it applies the standard Gaussian model of FTA,5 viz., n = pla1 P2a2 ... pa to a portion of itself, i.e., to the natural number exponents given by the Gaussian model, until the process terminates. Finiteness of the induction follows by the Well-ordering Principle. The final (unique) configuration of primes alone that represents a natural number is called a mosaic. For example, the mosaic of 10,000 is 222 522, and consists of primes alone. LEMMA 1 (alternate formulation of FTA). There exists a 1-1 effectively calculable function' v from the natural numbers onto mosaics (identify 1 with the "empty" mosaic). Upon taking the simple product of the primes alone appearing in the mosaic of a natural number, one can define an interesting effectively calculable number-theoretic function f,6 from the natural numbers onto the natural numbers.3 For example, recalling the mosaic of 10,000 given above, ,1 (10,000) = 2 2 2 2 2 5 = 160, and ,1 (s) = s for every square-free natural number s. LEMMA 2. Let N be the set of all natural numbers, i.e., strictly positive integers. Put 12 = {(+(.)), and define /t, k > 3, recursively. Then { 1: iEN is an infinite recursive set of effectively calculable number-theoretic functions, all intimately related to FTA. The proof of Lemma 2 depends, in part, upon S. C. Kleene's Basic Theorem concerning infinite recursive sets.7 At this point we introduce additive notions into the context of FTA in the obvious way, viz., defining another integer-valued number-theoretic function At* by taking the simple sum of the primes alone that appear in the mosaic of a natural number. For example, V,/*(10,000) = 2 + 2 + 2 + 2 + 2 + 5 = 15, and 4,*(p) = pfor every prime p. LEMMA 3. 4,t* is an effectively calculable number-theoretic function. At* (n) <