A seminatural system is a set N together with a binary relation < on N which A l . N is simply ordered with respect to <. A2. N is not the empty set.
A3.
If x E N, there exists y E N such that x < y.
By T ( x ) we mean ( y l y E N and y < x}.
It can be shown that N is well-ordered with respect to < . Binary operations of addition and multiplication can be defined on N which are associative and distributive, but not necessarily commutative [2]. Properties of the seminatural numbers under these operations are discussed in [2].
The purpose of this paper is the development of a number theory for the seminatural numbers, based on the properties exhibited in [2]. Such concepts as divisibility? greatest common factor, totient, prime number, and congruence will be defined and theorems analogous to those of conventional number theory such as the division and Euclidean aIgorithms, fundamental theorem of arithmetic, infinitude of primes, and EULER'S, FERMAT'S, and WILSOX'S theorems will be proved for the seminaturals.
satisfies the following axioms: [2]
Theorems referred to with numbers less than (32) appear in [2].
(32) Definition. A seminatural y is said to reach a seminatural x if there exists a finite set of seminaturals x,? x2? . . ., x, such that S y = x,, Xx, = x 2 , . . ., (33) Theorem. I f x is not primary, then there exists exactly one primary p such that p < x and such that p < q < x for 1u) primary q.
Proof. Let W be the set of all y which reach x. Since x = Sx', x' reaches x and W is not empty. Since the seminatural system N is well-ordered with respect to the relation < , W has a first element yo, which reaches x. Now yo is primary, for if yo = Sy', then y' would reach x and y' < yo. Suppose for some primary q , yo < q < x. Now there is a set xl, x2,. . ., x,, where S y , = xl, S x , = x 2 , . . ., XX,..., = x,, = x . By (2) (iv), x1 = Xy, < q , x2 = Sxl < q, . . ., x = S X , ~-~ < q , a contradiction. Hence yo is the desired primary p . That p is unique is evident.