Lesson plan
Divisibility sequence
In mathematics, a divisibility sequence is an integer sequence (a_n) indexed by positive integers such that : \text{if }m\mid n\text{ then }a_m\mid a_n for all and . That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined. A strong divisibility sequence is an integer sequence (a_n) such that for all positive integers and , : \gcd(a_m,a_n) = a_{\gcd(m,n)}, where is the greatest common divisor function. Every strong divisibility sequence is a divisibility sequence: \gcd(m,n) = m if and only if m\mid n . Therefore, by the strong divisibility property, \gcd(a_m,a_n) = a_m and therefore a_m\mid a_n .
Article summary
In mathematics, a divisibility sequence is an integer sequence (a_n) indexed by positive integers such that : \text{if }m\mid n\text{ then }a_m\mid a_n for all and . That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined. A strong divisibility sequence is an integer sequence (a_n) such that for all positive integers and , : \gcd(a_m,a_n) = a_{\gcd(m,n)}, where is the greatest common divisor function. Every strong divisibility sequence is a divisibility sequence: \gcd(m,n) = m if and only if m\mid n . Therefore, by the strong divisibility property, \gcd(a_m,a_n) = a_m and therefore a_m\mid a_n .
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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