%0 Journal Article
%T Redefining the Shape of Numbers and Three Forms of Calculation
%A Ji Peng
%J Open Access Library Journal
%V 8
%N 3
%P 1-22
%@ 2333-9721
%D 2021
%I Open Access Library
%R 10.4236/oalib.1107277
%X This paper redefines the Shape of numbers, makes it more natural and concise, and the domain of definition is extended to ring. The inconvenient PCHG() and PH() are removed. The concept of subsets is also removed. The new definition can be used to calculate ¡Æ<sub>n-0</sub><sup style="margin-left:-17px;">N-1</sup>¦°<sub>i-1</sub><sup style="margin-left:-15px;">M</sup> (K<sub>i</sub> n¡ÁD<i>i</i>)<br />
¡Æ<sub>n<sub>i,j</sub>-0</sub><sup style="margin-left:-21px;">j-N-1</sup>¦°<sub>i-1</sub><sup style="margin-left:-15px;">M</sup> (K<sub>i</sub> n<sub>i,j</sub>¡ÁD<i>i</i>), n<sub>i,j</sub>¡Ün<sub>i 1,j</sub> or n<sub>i,j</sub>=n<sub>i 1,j</sub>; K<sub>i</sub>,D<sub>i</sub>¡Êring. Three forms corresponding to three calculation methods are obtained. They can be used as a powerful tool for analysis. Some of the conclusions are: 1) Expressions and properties of two kinds of Stirling number, Lah number and Eulerian number; 2) Expression of power sum of natural numbers; 3) Vandermonde identity, Norlund identity; 4) New congruence and new proof of Wilson theorem; 5) ¡Æ<sub>n-1</sub><sup style="margin-left:-17px;">P-1</sup>¡Ô0 <i>MOD</i> <i>P</i><sup>2</sup>, P&gt;3; 6) ¡Æ<sub>C-0</sub><sup style="margin-left:-17px;">C-M-1</sup>(-1)<sup>M-1-C</sup>¡Æ<sub>PM(PS)-M,PB(PS)-C</sub><i>MIN(PS)</i>=1.
%K Shape of Numbers
%K Calculation Formula
%K Combinatorics
%K Congruence
%K Stirling Number
%K Eulerian Number
%U http://www.oalib.com/paper/6526651