I ran my latest prime-testing program, based on a cubic Frobenius test, on the numbers from 1 to $10^{11}$. Numbers in that range coprime to 30 are considered for testing. Given the input $n$, the program looks for the first polynomial $f_{i}$ such that $f_{i}$ is irreducible over $F_{n}$ , where $f_{i}$ is one of the 23 polynomials listed at the end of the earlier post https://dbernier.ca/2024/11/10/what-is-a-cubic-frobenius-primality-test/. Irreducibility is determined by table lookup. Occasionally, no polynomial is irreducible over $F_{n}$. Most often, in that case $n$ is composite although there are a few rare primes. Once the polynomial is selected, three congruences modulo $n$ are tested. We can represent any test result by a 3-bit binary sequence where a 1 bit means that a congruence is satisfied and a 0 bit means the congruence is not satisfied. The possible test results are thus $000, 001, 010, 011, 100, 101, 110, 111$ which can be read as an integer from 0 to 7. Then the summary statistics are as follows:
Type 0: 22548610645
Type 1: 16
Type 2: 154
Type 3: 0
Type 4: 4
Type 5: 0
Type 6: 0
Type 7: 4118054810
Grand total of tested and untested numbers between 1 and 100000000000: 26666666666
Grand total of probable primes found and untested primes, coprime to 30, between 1 and 100000000000: 4118054810
There are 22,548,610,645 results of type 0 where none of the three conditions is satisfied. There are 4,118,054,810 results of type 7 where all three conditions are satisfied. This is 3 less than $\pi(10^{11})=4,118,054,813$ because the primes 2, 3 and 5 (not being coprime to 30) are not even considered for testing. There are 154 results of type 2, 16 of type 1 and 4 of type 4 (these three types correspond to 1 of 3 conditions being satisfied). For types 3, 5 and 6 there are no test results (these types correspond to 2 of 3 conditions being satisfied). I might modify the program to show how many numbers with a type 7 result are composite, although this might slow down the program. The same primality test has been conducted for numbers up to $10^{13}$ with no pseudoprimes found (ie type 7 results for a composite input). A recent version of the program is available on Github here: https://github.com/mariotrevi/cubicFrobenius. Below, we have the program’s output relating to composites with test results of 1 through 6. But first, let us provide the summary statistics to $10^{10}$ for an updated version of the program that counts how many numbers with a type 7 result are prime, and that shows the prime count according to Kim Walisch’s primesieve (the job took 63 minutes): (and secondly, the summary statistics to $10^{11}$, a job that took 24 hours on my PC)
Composite and prime counts by type between 1 and 10000000000:
Type 0: 2211613577
Type 1: 13
Type 2: 83
Type 3: 0
Type 4: 4
Type 5: 0
Type 6: 0
Type 7: 455052508
Type 7 primes: 455052508
Validating Type 7 primes using Primesieve:
primesieve prime count: 455052511
Composite and prime counts by type between 1 and 100000000000:
Type 0: 22548610645
Type 1: 16
Type 2: 154
Type 3: 0
Type 4: 4
Type 5: 0
Type 6: 0
Type 7: 4118054810
Type 7 primes: 4118054810
Validating Type 7 primes using Primesieve:
primesieve prime count: 4118054813
Processing range: 1 to 100000000000
1243 test_type=1
3059 test_type=2
289 test_type=2
6859 test_type=2
2197 test_type=2
37249 test_type=1
50653 test_type=2
106597 test_type=2
237169 test_type=2
465283 test_type=2
573067 test_type=2
912673 test_type=2
935089 test_type=1
2813177 test_type=4
3464161 test_type=2
4330747 test_type=2
4826809 test_type=2
5603333 test_type=2
8025641 test_type=2
10991701 test_type=2
14687681 test_type=2
16636201 test_type=1
18210401 test_type=2
24242261 test_type=2
25916687 test_type=2
34173553 test_type=1
36487547 test_type=4
39866533 test_type=2
44482901 test_type=2
47045881 test_type=2
45407909 test_type=1
57833861 test_type=2
67194401 test_type=2
69348869 test_type=1
83403041 test_type=2
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103253333 test_type=2
109393201 test_type=2
119145439 test_type=2
136813753 test_type=2
140558671 test_type=2
140400719 test_type=1
159087769 test_type=4
176304427 test_type=4
238244041 test_type=2
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