https://github.com/abhrankan-chakrabarti/pari-gp-scripts
Bash wrappers for PARI/GP exploring classical number theory problems — Riemann zeta zeros and Gilbreath's conjecture.
https://github.com/abhrankan-chakrabarti/pari-gp-scripts
bash gilbreath-conjecture mathematics number-theory pari-gp riemann-zeta
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Bash wrappers for PARI/GP exploring classical number theory problems — Riemann zeta zeros and Gilbreath's conjecture.
- Host: GitHub
- URL: https://github.com/abhrankan-chakrabarti/pari-gp-scripts
- Owner: Abhrankan-Chakrabarti
- License: mit
- Created: 2026-06-13T21:47:39.000Z (4 days ago)
- Default Branch: main
- Last Pushed: 2026-06-14T09:38:50.000Z (3 days ago)
- Last Synced: 2026-06-14T11:15:55.943Z (3 days ago)
- Topics: bash, gilbreath-conjecture, mathematics, number-theory, pari-gp, riemann-zeta
- Language: Shell
- Homepage: http://abhrankan.dpdns.org/pari-gp-scripts/
- Size: 24.4 KB
- Stars: 1
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
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README
# PARI/GP Scripts
A collection of lightweight Bash wrappers for **PARI/GP** exploring classical problems in number theory.
---
## Table of Contents
- [Prerequisites](#prerequisites)
- [Installation](#installation)
- [Quick Start](#quick-start)
- [Scripts](#scripts)
- [riemann_zeros.sh](#riemann_zerossh)
- [gilbreath.sh](#gilbreathsh)
- [prime_gaps.sh](#prime_gapssh)
- [mobius.sh](#mobiussh)
- [Contributing](#contributing)
- [License](#license)
---
## Prerequisites
All scripts require **PARI/GP** (a computer algebra system specialized in number theory).
---
## Installation
* **Ubuntu / Debian:**
```bash
sudo apt-get update
sudo apt-get install pari-gp
```
* **macOS (via Homebrew):**
```bash
brew install pari
```
* **Fedora / RHEL:**
```bash
sudo dnf install pari-gp
```
* **Android (Termux):**
```bash
pkg install pari-gp
```
* **Windows:** Download the installer from [pari.math.u-bordeaux.fr](https://pari.math.u-bordeaux.fr/download.html)
---
## Quick Start
```bash
git clone https://github.com/Abhrankan-Chakrabarti/pari-gp-scripts.git
cd pari-gp-scripts
chmod +x riemann_zeros.sh gilbreath.sh prime_gaps.sh mobius.sh
./riemann_zeros.sh 10 30 38
./gilbreath.sh 10000 1
```
---
## Scripts
### riemann_zeros.sh
Computes the non-trivial zeros of the Riemann zeta function ζ(s) within a user-defined height range [n1, n2] on the critical line (Re(s) = 1/2).
Because non-trivial zeros always appear in complex conjugate pairs due to the symmetry of the zeta function (ζ(s̄) = ζ(s)‾), the output formats them using `0.5 +/- t*I` notation.
**Usage:**
```bash
./riemann_zeros.sh [n1] [n2] [precision]
```
**Example:**
```text
$ ./riemann_zeros.sh 10 30 38
1: 0.5 +/- 14.134725141734693790457251983562470271*I
2: 0.5 +/- 21.022039638771554992628479593896902777*I
3: 0.5 +/- 25.010857580145688763213790992562821819*I
```
**How it works:**
1. Validates that `n1`, `n2`, and `precision` are positive integers and that `n1 <= n2`
2. Passes the parameters to a `gp -q` session
3. Uses `lfuncreate(1)` and `lfunzeros()` to compute the imaginary parts of the zeros, then maps each `t` to `0.5 +/- t*I`
---
### gilbreath.sh
Generates a Gilbreath Triangle for the first `n` primes and verifies **Gilbreath's conjecture** at each row.
Starting from the sequence of the first `n` primes, each subsequent row is formed by taking the absolute differences of consecutive elements. Gilbreath's conjecture states that the first element of every row (after row 0) is always 1. It has been verified for millions of primes but remains unproven.
**Usage:**
```bash
./gilbreath.sh [n] [quiet]
```
`quiet` is optional: `0` (default) prints every row, `1` suppresses row output and only prints the result and timing.
**Example:**
```text
$ ./gilbreath.sh 6
Row 0: [2, 3, 5, 7, 11, 13]
Row 1: [1, 2, 2, 4, 2]
Row 2: [1, 0, 2, 2]
Row 3: [1, 2, 0]
Row 4: [1, 2]
Row 5: [1]
Gilbreath holds for first 6 primes
Time: 0.001 s
```
**Quiet mode** (useful for large `n`):
```text
$ ./gilbreath.sh 5000 1
Gilbreath holds for first 5000 primes
Time: 7.988 s
```
**How it works:**
1. Validates that `n` is a positive integer
2. Passes the parameters to `gp -q` via `echo | gp -q`
3. Uses `primes(n)` to generate the initial row, then iterates absolute differences via `vector(#row-1, i, abs(row[i]-row[i+1]))`
4. Reports computation time via PARI/GP's `gettime()`, formatted to 3 decimal places
---
### prime_gaps.sh
Computes the maximum gap between consecutive primes for the first `n` primes. Uses a streaming loop rather than allocating a vector, so memory usage is O(1) and it scales to very large `n`.
**Usage:**
```bash
./prime_gaps.sh [n]
```
**Example:**
```text
$ ./prime_gaps.sh 1000
Total Primes Processed: 1000
Last Prime Reached: 7919
Max gap found: 34
Time elapsed: 0.012 s
```
**How it works:**
1. Validates that `n` is an integer ≥ 2
2. Iterates through primes using `prime(i)`, tracking `p_prev` and `p_curr` to compute each gap on the fly
3. Reports total primes processed, last prime reached, maximum gap found, and elapsed time
---
### mobius.sh
Computes the Möbius function μ(n) for all integers up to N.
μ(n) is defined as: 1 if n = 1; (−1)^k if n is a product of k distinct primes; 0 if n has a squared prime factor.
**Usage:**
```bash
./mobius.sh [N]
```
**Example:**
```text
$ ./mobius.sh 10
n μ(n)
1 1
2 -1
3 -1
4 0
5 -1
6 1
7 -1
8 0
9 0
10 1
Time: 0.000 s
```
**How it works:**
1. Validates that `N` is a positive integer
2. Runs an optimized single-array sieve: for each prime `p`, zeros all multiples of `p²` first, then flips the sign of all non-zero multiples of `p` — half the memory footprint of the two-array approach with the same correctness
3. Prints each value in an aligned table; Unicode header printed from bash to preserve the μ symbol; full table suppressed for N > 1000
---
## Contributing
Pull requests are welcome. Ideas for new scripts include:
- Goldbach partition checker
- Dirichlet L-function zeros
- Euler product approximations of π
Each script should follow the conventions of the existing ones:
- Bash wrapper with `gp` availability check
- Argument mode with interactive prompt fallback
- Input validation before passing to GP
- Consistent header comment block (Title, Description, Dependencies)
---
## License
This project is licensed under the MIT License — see the [LICENSE](LICENSE) file for details.