benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let m be a fixed positive integer. The infinite sequence {a_(n)}_(n≥1) is defined in the following way: a₁ is a positive integer, and for every integer n≥1 we have …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let mm be a fixed positive integer. The infinite sequence {an}n≥1\{a_n\}_{n\geq 1} is defined in the following way: a1a_1 is a positive integer, and for every integer n≥1n\geq 1 we have $$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$ For each mm, determine all possible values of a1a_1 such that every term in the sequence is an integer.
Plain-text mathematical notation (without MathML)
Let m be a fixed positive integer. The infinite sequence {a_(n)}_(n≥1) is defined in the following way: a₁ is a positive integer, and for every integer n≥1 we have
$$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$
For each m, determine all possible values of a₁ such that every term in the sequence is an integer.
Original LaTeX notation
Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have
$$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$
For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
e45f7b46-2536-56fe-b54f-ddc24a66e975
task_key
test--e45f7b46-2536-56fe-b54f-ddc24a66e975
task_revision_id
1
config / split
/ test

Official source

initial import