Mason

independent computational replication

Fibonacci residues modulo prime powers

I built this computational study to see which residues appear when Fibonacci-type sequences are reduced modulo p^k. I wrote the Python code, ran the experiments, checked the outputs, and documented the method myself.

Status and authorship: I independently implemented and completed this project. I worked through the method, wrote and tested the code, reproduced the results, and verified the saved outputs myself. The Fibonacci benchmarks are published mathematics, and the Lucas and Pell observations below are finite experiments, not proofs or claims of new discoveries.

what I tested

One number can repeat before the whole recurrence cycles, so stopping at a repeated value would be wrong. My program tracks pairs of consecutive values instead.

x(n + 2) = a x(n + 1) + b x(n) mod m
state(n) = (x(n), x(n + 1))

For each modulus m = p^k, my code follows the pair state and keeps the first coordinate from every step. It stops when the original pair comes back. I worked through and verified each of these choices before using them.

what I found

My dataset contains 147 runs across eight primes. The checks reproduced selected published Fibonacci benchmarks, and the runs showed:

• Fibonacci with p = 13 at 9/13 for the tested exponents k = 1 through 5.
• Lucas with p = 13 stayed at 12/13. With p = 5, it fell from 4/5 toward about one third.
• Pell at p = 13 fell from 9/13 to 1/13 at the first lift, then stayed at 1/13 through k = 5.
• Pell at p = 31 fell from 19/31 to 22/961, then stayed at 22/961 through k = 4.

a question from the results

My Pell runs produced different first lifts for 13 and 31, followed by matching growth in the state period and residue count. That led to a question for a wider, predeclared test range.

Question for future work: if the Pell state period and residue count both grow by p at one prime-power lift, does the residue proportion stay fixed at later non-exceptional lifts?

The tested cases motivate the question, but they do not prove it. I am not claiming the pattern as a new discovery.

files and checks

Every saved run returned to its starting pair. A separate small-modulus implementation checked 78 cases, and five unit tests cover known periods, residue counts, and edge cases. I reran these checks, audited the code, and compared the Fibonacci output with published benchmarks. Finite tests support the displayed data, but they do not prove an infinite pattern.

I wrote, ran, and verified this code and analysis independently. The downloads are the notebook, source, data, tests, and figures behind the page.