Hung-ping Tsao (2026). Combined Mathematics II: Talk Elegancy, Number Intricacy, Life Contingency, Sudoku Ecstacy, Padagogy Efficiency, Prime Normalcy and Math Sufficiency,. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Volume 8, Number 3A, March 2026; 415 pages. Lenox Institute Press MA, USA. ...... ABSTRACT: I am an amateur mathematician, who has been working rather diligently all his life. After receiving Ph.D. from UIUC in the area of combinatorics and teaching for two years in college, I pursued my actuarial career for eight years before returning to teach. During the 17 years of teaching at SFSU, I used the textbook "College Mathematics" tailor-made for my own students in College of Business, which has recently been ranked number 1 in https://www.academia.edu/Documents/Business Mathmatics/TopPapers. Ever since my retirement in the year of 2002, I have been working on Combinatorial Number Theory. Using only elementary methods, I was able to get some breakthroughs in powered sums and triangular arrays of numbers. I hope my efforts and results could evolve into an undergraduate textbook in the area of Combinatorial Number Theory. All of the above efforts and results had been included in "Combined Mathematics" and highly appreciated worldwide since its publication a year ago. This sequal of "Combined Mathematics" will maintain its original integrity except for some minor corrections and the addition of page numbers of items in Table of Contents, suggested by my highschool classmate Professor Tsu-Wei Chou. In terms of the terminal value sequence t(n), we can define the second terminal value sequence ts(n) via t0(n)=n, t1(n)=t0(n), t2(n)=t(t0(n)+t1(n)), t3(n)=t(t0(n)+t1(n)+t2(n)), … and introduce the terminal value matrix ts(n) and the third terminal value sequence Ts(n) via T0(n)=t0(n),T1(n)=t1(n), T2(n)=t(T1(t0(n)+t1(n))), T3(n)=t(T2(t1(n)+t2(n))), …and the terminal value matrix Tm(kxk) as well. Furthermore, we can combine ts(n) and Ts(n) to form the terminal value matroid tTs(n). We have found in 1 that the percetage of 5, 7, 11 and 13 in tm(60x60) to be .27, .13, .10 and .10 respectively, which are amazingly close to .27, .16, .06 and .07 for t(k) up to k=100000. In such a one-man kung fu jungle, for some 200 natural numbers, there are only 10 basic types, with 8 in the first 100 natural numbers, of ts and Ts, respectively. We believe that the percentages of small primes in tm(kxk) and t(kxk) will stay close but much smaller than those of Tm(kxk). Therefore, the distribution of small primes in the terminal value matroid tTms(n) are much denser in the third dimension than the first two. This makes me wonder: could this reflect any phenominon in our universe or outer spaces? As the sequal of Combined Mathematics, we shall add a section of Math Sufficiency for AI Era and attach an easier solution for the hardest Sudoku from 24 in APPENDIX B.
Tsao et al. (Sun,) studied this question.