You can deploy this model and start earning money today!

cahlen/
erdos-straus-cuda

Name: erdos-straus-cuda
Availability: InStock

computational-mathematics

cahlen/continued-fraction-spectra

mit

Views

No views yet

Erdos-Straus Solution Counter

GPU kernel for counting solutions $f(p)$ to the Erdos-Straus equation:

$$\frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$

for each prime $p$, where $x \leq y \leq z$.

The Erdos-Straus conjecture (1948) asserts that $f(p) \geq 1$ for all primes $p \geq 2$. This kernel computes the exact count of ordered solutions for every prime in a batch.

Usage

python

# /// script
# dependencies = ["torch", "kernels"]
# ///
import torch
from kernels import get_kernel

erdos_straus = get_kernel("cahlen/erdos-straus-cuda")

# Count solutions for a batch of primes
primes = torch.tensor([2, 3, 5, 7, 11, 13, 97, 9973], dtype=torch.int64, device="cuda")
counts = erdos_straus.count(primes)
print(dict(zip(primes.tolist(), counts.tolist())))
# {2: 1, 3: 3, 5: 2, 7: 5, 11: 8, 13: 12, 97: 251, 9973: 62624}

API

`erdos_straus.count(primes: Tensor) -> Tensor`

Parameter	Type	Description
`primes`	`Tensor[int64]` (1-D, CUDA)	Batch of primes to count solutions for
Returns	`Tensor[int32]` (1-D, CUDA)	Solution count $f(p)$ for each prime

Algorithm

For each prime $p$, the kernel enumerates all valid $(x, y, z)$ triples:

Iterate $x$ from $\lceil p/4 \rceil + 1$ to $\lfloor 3p/4 \rfloor$
Compute the remainder fraction $\frac{4x - p}{px}$ and iterate $y$ in the valid range
Check if $z = \frac{px \cdot y}{(4x-p)y - px}$ is a positive integer with $z \geq y$

Each thread handles one prime. Throughput scales linearly with GPU cores.

Results

Verified on 8x NVIDIA B200 GPUs at bigcompute.science:

10^8: 5.76M primes, conjecture holds (all $f(p) \geq 1$)
10^9: 50.8M primes, in progress

All data open at github.com/cahlen/idontknow.

Citation

bibtex

@misc{humphreys2026erdosstraus,
  author = {Humphreys, Cahlen},
  title = {GPU-Accelerated Erdos-Straus Solution Counting},
  year = {2026},
  url = {https://bigcompute.science}
}

Human-AI collaborative research. Not peer-reviewed. All code and data open for verification.