Omega Divisibility Tests for Bases 2 ≤ r ≤ 93

Omega Divisibility Tests for Number Bases 2 through 93

This table “Prime Composition versus Divisibility Rules” proved seminal in my understanding of the importance in integer base r of its last digit, omega ω = (r − 1). The primes populate the horizontal axis, the integers 2 ≤ r ≤ 93 the vertical axis. The multiplicities of primes in the prime decomposition of r are listed in the data for each base r. The presence of the divisors of ω in base r shadow the multiplicity of the divisor in base ω. This page evolved into the unpublished paper “Phantom Divisibility”. Eventually I recognized an alpha relationship, i.e., the relationship of α = (r + 1), thus the neighbor relationship of α and ω with r. The neighbor-related or indirect relationship merged with thoughts on the direct relationship of digits with their bases, leading to the digit map and spectrum concept of late 2009 expressed in the 2 Mb PDF “Digit Base Relationship” d87a9. This evolved into the expanded “Digit Base Relationship” presentation, 11Mb d87aa. The indirect relationships have grown to become a dominant set of intuitive methods of manipulation of number, expressed in the 2011 paper “Dozenal FAQs”, 5.6Mb PDF d8907, and incorporated into a peer-reviewed article “Exploring Number Bases as Tools” published in the March 2012 edition of ACM-Inroads, a magazine of computer science education. Page written 9 July 2007, tayya 79aa, seven dozen ninth phase (Salcyra-Karlmelal Xrga, “Life Phase of Karl Michael, Son”), St. Louis.

This page last modified Thursday 12 April 2012.