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OlympiadBench / 1631 / For a positive integer a, define a sequence of integers x₁,x₂,……

Problem

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

question

For a positive integer aa, define a sequence of integers x1,x2,…x_{1}, x_{2}, \ldots by letting x1=ax_{1}=a and xn+1=2xn+1x_{n+1}=2 x_{n}+1 for n≥1n \geq 1. Let yn=2xn−1y_{n}=2^{x_{n}}-1. Determine the largest possible kk such that, for some positive integer aa, the numbers y1,…,yky_{1}, \ldots, y_{k} are all prime.
Plain-text mathematical notation (without MathML)
For a positive integer a, define a sequence of integers x₁,x₂,… by letting x₁=a and x_(n+1)=2x_(n)+1 for n≥1. Let y_(n)=2^(x_(n))−1. Determine the largest possible k such that, for some positive integer a, the numbers y₁,…,y_(k) are all prime.
Original LaTeX notation
For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \ldots$ by letting $x_{1}=a$ and $x_{n+1}=2 x_{n}+1$ for $n \geq 1$. Let $y_{n}=2^{x_{n}}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_{1}, \ldots, y_{k}$ are all prime.

answer type

Numerical

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID
1631
task_id
273cf6cd-be5d-51a6-806c-29a475905f3f
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1631
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import