Pisano period
E337574
The Pisano period is the repeating cycle length of Fibonacci numbers when taken modulo a given integer.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Pisano period canonical | 1 |
| Pisano period modulo n | 1 |
| the Pisano period π(n) is the smallest positive integer k such that F_{m+k} ≡ F_m (mod n) for all m | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3214205 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Pisano period Context triple: [Fibonacci sequence, moduloPatternName, Pisano period]
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A.
Fibonacci sequence
The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
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B.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
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C.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
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D.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
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E.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Pisano period Target entity description: The Pisano period is the repeating cycle length of Fibonacci numbers when taken modulo a given integer.
-
A.
Fibonacci sequence
The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
-
B.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
C.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
D.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
E.
Fermat's little theorem
Fermat's little theorem is a fundamental result in number theory that characterizes how prime numbers interact with integer powers modulo that prime, forming the basis for many modern cryptographic algorithms.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
number theory concept ⓘ |
| alsoKnownAs |
Pisano period modulo n
ⓘ
linked to:
Pisano period
|
| appearsIn |
literature on Fibonacci numbers and their applications
ⓘ
research on periodicity of linear recurrences modulo integers ⓘ |
| computableBy | iterating Fibonacci numbers modulo n until the pair (0,1) reappears ⓘ |
| definition | the length of the repeating cycle of Fibonacci numbers taken modulo a given integer n ⓘ |
| dependsOn |
Fibonacci sequence
ⓘ
modular arithmetic ⓘ |
| domain | positive integers n ≥ 1 ⓘ |
| field | number theory ⓘ |
| namedAfter |
Leonardo Pisano
ⓘ
linked to:
Leonardo Fibonacci
|
| property |
for each integer n ≥ 1, the Fibonacci sequence modulo n is periodic
ⓘ
for prime p, π(p) divides p − 1, 2(p + 1), or 2(2p + 2) depending on p modulo 10, with exceptions ⓘ if m divides n then π(m) divides lcm(π(n), m) in many cases ⓘ the Pisano period π(n) is finite for every positive integer n ⓘ the Pisano period π(n) is the period of the pair (F_k, F_{k+1}) modulo n ⓘ the Pisano period π(n) is the smallest positive integer k such that F_{m+k} ≡ F_m (mod n) for all m ⓘ
linked to:
Pisano period
the sequence of Fibonacci numbers modulo n always eventually repeats with initial pair (0,1) ⓘ π(1) = 1 ⓘ π(10) = 60 ⓘ π(11) = 10 ⓘ π(12) = 24 ⓘ π(13) = 28 ⓘ π(2) = 3 ⓘ π(20) = 60 ⓘ π(3) = 8 ⓘ π(4) = 6 ⓘ π(5) = 20 ⓘ π(6) = 24 ⓘ π(60) = 60 ⓘ π(7) = 16 ⓘ π(8) = 12 ⓘ π(9) = 24 ⓘ π(n) divides 2φ(n)·n for all n, where φ is Euler's totient function ⓘ π(n) divides 6n for all n ⓘ π(n) is submultiplicative in many cases, with π(mn) related to lcm(π(m), π(n)) when m and n are coprime ⓘ π(n) is the length of the cycle from the first occurrence of (0,1) to its next occurrence ⓘ |
| relatedTo |
Fibonacci numbers
ⓘ
linked to:
Fibonacci sequence
Fibonacci sequence modulo n ⓘ Lucas sequences ⓘ order of an element in a multiplicative group modulo n ⓘ period of linear recurrences modulo n ⓘ |
| symbol | π(n) ⓘ |
| usedIn |
analysis of algorithms involving Fibonacci numbers modulo n
ⓘ
cryptographic constructions involving Fibonacci-like sequences ⓘ pseudorandom number generation based on Fibonacci recurrences ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Pisano period Description of subject: The Pisano period is the repeating cycle length of Fibonacci numbers when taken modulo a given integer.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Pisano period
Pisano period
→
property
→
the Pisano period π(n) is the smallest positive integer k such that F_{m+k} ≡ F_m (mod n) for all m
ⓘ
linked to: Pisano period