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In this paper we present a new mathematical conception based on a new methodfor ordering the integers. The method relies on the assumption that negativenumbers are beyond infinity, which goes back to Wallis and Euler. We alsopresent a new axiom system, the model of which is arithmetics. We defineregular method for summation of infinite series which allows us to discovergeneral and unified approach to summation of divergent series, and determinethe limits of unbounded and oscillating functions. Several properties fordivergent series and explicit formulas for sums of some infinite series areestablished. A number of finite and new recurrence formulas for Bernoullinumbers are obtained. We rederive some known results, but in a simpler andelementary way, and establish new results by means of techniques of thetheoretical background developed.

On one number-theoretic conception: towards a new theory