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Omni-MATH / For which positive integers m does there exist an infinite arithmetic sequence of integers a₁,a₂,⋯ and an infinite geometric sequence of integers g₁,g₂,⋯ satisfying the following p…

Problem

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problem

For which positive integers

m

does there exist an infinite arithmetic sequence of integers

a_1,a_2,\cdots

and an infinite geometric sequence of integers

g_1,g_2,\cdots

satisfying the following properties?

\bullet

a_n-g_n

is divisible by

m

for all integers

n>1

;

\bullet

a_2-a_1

is not divisible by

m

Plain-text mathematical notation (without MathML)

For which positive integers m does there exist an infinite arithmetic sequence of integers a₁,a₂,⋯ and an infinite geometric sequence of integers g₁,g₂,⋯ satisfying the following properties?
•  a_(n)−g_(n) is divisible by m for all integers n>1 ;
•  a₂−a₁ is not divisible by m .

Original LaTeX notation

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1,a_2,\cdots$ and an infinite geometric sequence of integers $g_1,g_2,\cdots$ satisfying the following properties?
$\bullet$  $a_n-g_n$ is divisible by $m$ for all integers $n>1$ ;
$\bullet$  $a_2-a_1$ is not divisible by $m$ .

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Source and history

task_id: 3dc3c87e-01f9-57d6-8a85-112cc4339b09
task_key: test--3dc3c87e-01f9-57d6-8a85-112cc4339b09
task_revision_id: 2
config / split: / test

Official source

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