Pisano Period Calculator

Find Pisano periods for single or multiple moduli accurately today with confidence. Choose exact scanning, factorization assistance, or mixed verification for certainty and clarity. Batch mode processes ranges, exporting comprehensive tables for analysis and fast reporting. See repeating pairs, first return time, and sequence fragments clearly. Download CSV and PDF reports with one click.

Inputs

Tip: Exact scan runs in O(π(m)) steps. Factorization uses number-theoretic formulas and LCM over prime powers.

Example Data Table

Known small moduli and their Pisano periods.

m	π(m)	Notes
2	3	Special power of 2
3	8	Prime, 3 ≡ −2 (mod 5)
4	6	Power of 2
5	20	Base case for powers of 5
6	24	lcm(π(2), π(3))
7	16	Prime, 7 ≡ 2 (mod 5)
8	12	π(2^3) = 3·2^{2}
9	24	π(3^2) = 3·π(3)
10	60	lcm(π(2), π(5))
11	10	Prime, 11 ≡ 1 (mod 5)

Formula Used

The Fibonacci sequence modulo m is periodic; its period is π(m).
π(mn) = lcm(π(m), π(n)) when gcd(m, n) = 1.
For p an odd prime, π(p) divides p−1 if p ≡ ±1 (mod 5), else divides 2(p+1).
For prime powers: π(2) = 3, π(4) = 6, and π(2^k) = 3·2^k−1 for k ≥ 3.
π(5) = 20 and π(5^k) = 20·5^k−1.
For odd p ≠ 5: π(p^k) = π(p)·p^k−1.
Exact scan detects the first return of the pair (0,1) modulo m.

How to Use

Enter modulus m > 1. Optionally add batch values or ranges.
Select Exact scan for guaranteed results; choose Factorization for speed.
Pick Both to cross‑check and highlight any discrepancies.
Enable sequence display to see the pair cycle up to one period.
Export any table as CSV or PDF using the buttons provided.

FAQs

It is the length of the repeating cycle of Fibonacci numbers taken modulo m, i.e., the smallest k > 0 with (F_k, F_k+1) ≡ (0,1) (mod m).

It leverages known number‑theoretic formulas for primes and prime powers and combines them via least common multiples, typically running much faster than scanning.

Disagreement usually indicates a step limit in scanning or an incomplete factorization for very large m. Increase the limit or use both modes to verify.

Scanning needs O(π(m)) steps, about O(m) on average. Factorization handles larger m, though trial division has limits; moderate sizes work well.

Yes. Toggle sequence display, then use the CSV or PDF buttons beneath the sequence grid to export the shown cycle pairs.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.

Inputs

Example Data Table

Formula Used

How to Use

FAQs

What is a Pisano period?

Why offer factorization mode?

When could scan and factorization disagree?

How large a modulus is practical?

Can I export the sequence itself?

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