Recently, I’ve been studying the 2003 paper of Chen and Greene “Some Comments on Baillie-PSW Pseudoprimes” published in the Fibonnacci Quarterly in 2003. I wish to make a record of my thinking about this problem, which is occasionally discussed online, for instance in this Mathematics Stack Exchange discssion thread: https://math.stackexchange.com/questions/2902592/isnt-the-estimate-for-the-smallest-counterexample-of-the-bpsw-test-far-too-opti . The problem studied by… Continue reading On Baillie-PSW pseudoprimes following Chen and Greene
Tag: Mathematics
Conjecture Verified in Lean for q<57000
After more than a month of working with Aristotle of Harmonic, I’m reasonably confident that we’ve shown either in Lean or in Pari/gp enough to justify the statement: Let q<57000 be an odd prime such that 4q=a^2+27 for some positive integer a. Let N be an odd prime such that N=/=q and gcd(a,N)=1. Let s1… Continue reading Conjecture Verified in Lean for q<57000
Empirical Results on a Randomized Cubic Primality Test
In 1980, Baillie and Wagstaff proposed a novel primality test consisting of a strong Fermat base $2$ test and a standard or strong Lucas test (Baillie and Wagstaff, 1980). This test has come to be known as the Baillie-PSW test after the names of its creators, and is now commonly included in software for mathematics,… Continue reading Empirical Results on a Randomized Cubic Primality Test
Pari/gp code 12/28/2025
Implementation of the Randomized Cubic Primality Test in Python
This post provides a complete, runnable Python implementation of the randomized cubic primality test using explicit Frobenius computation in a cubic ring, as described in my article “A Randomized Cubic Primality Test Using Explicit Frobenius Computation”. The test operates in the ring $$R_N = (\mathbb{Z}/N\mathbb{Z})[x]/(f(x)),\qquad f(x)=x^3-qx-q,$$ where $N$ is the integer under test and $q$… Continue reading Implementation of the Randomized Cubic Primality Test in Python
The Feigenbaum alpha Constant to 16,000 decimals
A Challenge in Computational Number Theory: Finding Cubic Frobenius Pseudoprimes
A Primality Test Based on Vieta’s Formulas
Introduction Primality testing is of fundamental importance to number theory. For numbers $n$ up to about a million,high-school level methods such as trial division by the integers $d$ such that $d \leq \sqrt n$ are simpleand effective. However, they are no match even for $30$-digit numbers. Let us consider the $32$-digit number $n = 10000000000000000000000000000033$.… Continue reading A Primality Test Based on Vieta’s Formulas
A computation of test passers to 40 billion
I had an AI optimize my C program that finds the numbers that pass the first criterion of the test mentioned in the numbers game blog post of April 26 (in a specified range). I tested the various versions repeatedly and compared the output of passers (numbers passing the 1st criterion) with files of bona… Continue reading A computation of test passers to 40 billion
Python script to test random numbers
I had an AI write a Python program to test random numbers in the numbers game from my last post. On the command line, one gives two numerical arguments: the number of iterations and the upper limit on the numbers. So python3 vietatest.py 20 1000000000000 will run 20 tests on random numbers in the range… Continue reading Python script to test random numbers
Numbers game Python code for Beta testers
I’m releasing beta code for a numbers game written in Python. The code was written by the Chatbot ChatGPT. Update: I’ve added a C language program after the Python program. The program interactively solicits the user to enter non-negative integers. Numbers with a hundred or more digits are acceptable. I don’t know how to run… Continue reading Numbers game Python code for Beta testers
Cubic Vieta-Frobenius test
DeepSeek AI’s reply to my math question in real analysis
Given a continuous, unbounded function ( f: \mathbb{R} \to \mathbb{R} ), we need to determine if there exists an ( x \in \mathbb{R} ) such that the set ( { |f(nx)| \, | \, n \in \mathbb{Z} } ) is unbounded. Thus, the answer is (\boxed{Yes}).
A finite fields probable prime test using Galois conjugates and Vieta’s formulas
I’ve been trying to extend the cubic Frobenius test to a quartic test, and it’s been challenging. I got an idea for a primality test that isn’t fast, but is interesting in any case. Let $n$ be a probable prime, and $f$ be a monic 4th degree polynomial such that $f$ has no roots in… Continue reading A finite fields probable prime test using Galois conjugates and Vieta’s formulas
Cubic Frobenius test statistics from 1.0e13 to 2.0e13
For composites in the range 1.0e13 to 2.0e13, there are two instances of type 2 (passing Congruence 1 only) and no instances showing they passed either Congruence 2 or Congruence 3. For the primes in the range, they all pass congruences 1 to 5. Processing block 1000: 10999000000000 to 11000000000000Type 0: 233313317052Type 1: 0Type 2:… Continue reading Cubic Frobenius test statistics from 1.0e13 to 2.0e13
When might one expect a Fermat-Lucas pseudoprime?
The Baillie-PSW primality test consists of a strong Fermat base $2$ test and a standard or strong Lucas sequences test. For an odd number $n$, the Fermat base $2$ test consists in checking whether the congruence $2^{n-1} \equiv 1 \pmod n$. A composite $n$ that passes the Fermat base $2$ test is known as a… Continue reading When might one expect a Fermat-Lucas pseudoprime?
Random subset heuristics applied to pseudoprimes
Suppose there are N tickets in an urn of which m < N are winning tickets. You have n tries to get a winning ticket. The tickets are replacedin the urn after each draw. Assume m and n are much smaller than N. Show that the probability of failure p_fail is approximated by: p_fail ~=… Continue reading Random subset heuristics applied to pseudoprimes
A naive implementation of Erdos’ Carmichael number construction
I’ve been reading some about the Erdos heuristic to construct Carmichael numbers, first described by Erdos in 1956. It is based on Korselt’s criterion, that N is a Carmichael number if N is composite, squarefree and such that if p|N, then (p-1)|(N-1). The best reference I’ve found to understand the basics of Erdos’s heuristic is… Continue reading A naive implementation of Erdos’ Carmichael number construction
The cubic Frobenius primality test as a Pari/gp script
Cubic Frobenius test statistics to $10^{12}$
The test considers numbers (greater than 1) coprime to 30 in a given range. If the test number is denoted by $n$, we assume $n$ is prime and try to prove by contradiction that it is composite, i.e. that it fails one or more of the five congruences described in an earlier blog post (https://atomic-temporary-23414054.wpcomstaging.com/2024/11/13/a-perrin-like-primality-test-with-three-congruences/.The)… Continue reading Cubic Frobenius test statistics to $10^{12}$
A Conjecture for a Perrin-like sequence
Conjecture Let $ W(k) $ be defined as:$$W(0) = 1, \quad W(1) = 0, \quad W(2) = 2,$$and for $ k > 2 $, recursively by:$$W(k) = 3W(k-2) + W(k-3).$$ Then, for $ n > 0 $ such that $ n \equiv \pm 2, \pm 4 \pmod{9} $, $ n $ is prime if and… Continue reading A Conjecture for a Perrin-like sequence
Is ChatGPT getting smarter? (maybe)
I wrote: A typical (even math major) college undergraduate couldn’t do this. AI might be approaching graduate level in math in some respects… ChatGPT replied: You’re absolutely right—this problem requires a deep understanding of analysis and topology, tools like the Baire Category Theorem, and the ability to apply them creatively in a non-obvious way. It’s… Continue reading Is ChatGPT getting smarter? (maybe)
Cubic Frobenius test statistics to $10^{11}$
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… Continue reading Cubic Frobenius test statistics to $10^{11}$
New repository on Github for trying out my code
I’ve created a software repository on Github hosting the files in my project: cubicFrobenius. Here are some commands in Linux bash that you may find helpful:
Sample output from my latest prime-testing program
Types 1, 2 and 4 each correspond to one of three conditions being met. Type 0 corresponds to no condition being met, while type 7 corresponds to 3 out of 3 conditions being met. Some numbers $n$ are reported as untested because none of the 23 polynomials is irreducible over $\mathbb{F}_{n}$. The data on numbers… Continue reading Sample output from my latest prime-testing program
Detailed prime test statistics
Below, we have a sample report using the most up to date version of the program:
cos(z)=0 implies z is real // ChatGPT’s solution
To determine whether $ \cos(z) = 0 $ implies that $ z $ must be real, we analyze the equation using the definition of cosine in the complex domain. Key Idea: Using the Complex Definition of Cosine The cosine function for a complex number $ z = x + iy $ (where $ x, y… Continue reading cos(z)=0 implies z is real // ChatGPT’s solution
A Perrin-like Primality Test with Five Congruences
The Perrin sequence is defined by $P(0)=3$, $P(1)=0$, $P(2)=2$ and the recurrence relation $P(n) = P(n-2) + P(n-3)$ for $n\gt 2$. The Perrin sequence has the property that, if $n$ is prime then $n|P(n)$. However, the converse is false: in 1982, Adams and Shanks showed that $271441|P(271441)$, where $271441 = 521^2$. In contrast to the… Continue reading A Perrin-like Primality Test with Five Congruences
Update on the cubic Frobenius test
I’m now using the polynomial $f = X^3 – 3X -1$. Given a number to test $p$, things are good if $f$ is irreducible over $F_{p}$. I’m not too sure why this is so, but it’s also the case for the standard quadratic Frobenius test, that one works in proper extensions of $F_{p}$. Empirically, the… Continue reading Update on the cubic Frobenius test
Test 4
Certainly! Below is the LaTeX-formatted memorandum tailored for your WordPress blog. This version avoids using double $$ signs for display math and instead utilizes \[ … \] for display equations and $…$ for inline math, ensuring compatibility with most WordPress math rendering plugins like MathJax or KaTeX. Enhancing the Frobenius Primality Test Using Cubic Recurrence… Continue reading Test 4
Latex test 3
Certainly! Below is the LaTeX-formatted version of the memorandum tailored for your WordPress blog. This version avoids using double $$ signs for display math and instead utilizes single $ signs for all mathematical expressions, ensuring compatibility with WordPress’s math rendering plugins. Enhancing the Frobenius Primality Test Using Cubic Recurrence Relations Introduction In the pursuit of… Continue reading Latex test 3
Symmetric polynomials and Galois Theory applied to primality testing (work in progress)
Memorandum: Leveraging Symmetric Polynomials and Galois Theory for Primality Testing To: Mathematics Enthusiasts and ScholarsFrom: [Your Name]Date: [Current Date]Subject: Theoretical Framework for Primality Testing Using Symmetric Polynomials and Galois Theory In the quest to develop robust primality tests, leveraging foundational algebraic structures offers profound insights and efficient methodologies. This memorandum elucidates a theoretical framework that… Continue reading Symmetric polynomials and Galois Theory applied to primality testing (work in progress)
A result in real analysis as summarized by ChatGPT-4
Note: I had a long discussion with ChatGPT over the proof of a result in real analysis at the grad school level. I coaxed it and guided it into seemingly understanding the details of the argument. Then, I asked it to summarize the result and its proof. What follow is its response to my prompt.… Continue reading A result in real analysis as summarized by ChatGPT-4