If our goal were to find prime numbers, it is easy to show that using the matrix and seed combination is more productive than not using it. Assuming we were to test a Reduced Set of all numbers that end in 1,3,7 or 9 for primeness, then we could say 40% of all the numbers between a lower prime factorial PF(x) and upper of PF(x+1) would need to be tested. Using the classic approximation of prime numbers formula y/(Log(y)-1) for the count of prime numbers smaller than y, we can find the approximate count of primes that fall between the lower and upper PF numbers. Thus we can chart the approximate density of primes as a percent of numbers in the Reduced Set in each PF interval. Below you can see that fraction plotted in green and starting with Matrix 6. It declines from 20% to about 2% by the 29th Matrix. Similarly I show the percent of all numbers that belong to the Matrix(x) when x varies from 6 to 29 (line shown in blue). By the 29th matrix, it is still declining but at a very slow rate and it appears that it might never decline below 10%. The red line is the density of prime numbers in the matrix. By the 29th matrix, that percentage has declined to near 8%. However, the rate of decline from one matrix to the next is less than the rate of decline of the green line. This is an important fact because it means that as the matrix number x gets larger, the efficiency of the matrix computation is improving relative to examining all numbers ending in 1,3,7 and 9.
This is further suggested mathematically by knowing that each subsequent matrix is reduced by the number with the most multiples in the matrix, the Key Seed. I suspect but have not confirmed that the rate of efficiency is tied to the rate at which the ratio of average prime to prior prime is also decreasing and asymptotically approaches 1.
Matrix Efficiency Ratio of Finding Primes
The efficiency ratio of the Matrix in finding prime numbers can be determined by dividing the density of prime numbers in the Matrix by the density of prime numbers in the Reduced Set. We can directly compute the exact size of each matrix and we also know the number of numbers between each PF number and multiplying that by .4 gives us the exact number of numbers in the Reduced Set. The number of primes in this interval is also known by estimation. We can compute the density of primes in both of these sets. Then we can compute the ratio of the density of primes in the Matrix to the density of primes in the Reduced Set. This is the same as dividing the number of numbers in the Reduced Set by the number of numbers in the Matrix since the prime number count is identical in the numerators of the density ratios and they cancel out. The results are shown in the graph below.
We can see that at the Matrix 3 level the matrix is 1.5 times as dense with prime numbers as the Reduced Set. It gets progressively more efficient as the matrix size increases and reaches 3.5 by about Matrix 30 which has a largest number of about 48 digits. The efficiency increases more slowly as the matrix size increases. That fact combined with the ease with which matrix members can be computed supports the idea that using matrix computation is a shortcut to discovering sets of large numbers where the probability of being prime is increased.