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Omni-MATH / Let u be a positive rational number and m be a positive integer. Define a sequence q_1,q_2,q_3,dotsc such that q₁=u and for n≥2: …

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Let uu be a positive rational number and mm be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that q1=uq_1=u and for n≥2n\geqslant 2: if qn−1=ab for some relatively prime positive integers a and b, then qn=a+mbb+1.\text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}. Determine all positive integers mm such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number uu.
Plain-text mathematical notation (without MathML)
Let u be a positive rational number and m be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that q₁=u and for n≥2:
if q_(n−1)=(a)/(b) for some relatively prime positive integers a and b, then q_(n)=(a+mb)/(b+1). 
Determine all positive integers m such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number u.
Original LaTeX notation
Let $u$ be a positive rational number and $m$ be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that $q_1=u$ and for $n\geqslant 2$:
$$\text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}.$$ 
Determine all positive integers $m$ such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number $u$.

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task_id
fefecc19-b7c2-5002-a9d3-dcfec6745796
task_key
test--fefecc19-b7c2-5002-a9d3-dcfec6745796
task_revision_id
1
config / split
/ test

Official source

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