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Omni-MATH / Let a₁,a₂,a₃,… be a sequence of positive integers where a₁=∑_(i=0)¹⁰⁰i! and a_(i)+a_(i+1) is an odd perfect square for all i≥1. Compute the smallest possible value of a₁₀₀₀.

Problem

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problem

Let a1,a2,a3,…a_{1}, a_{2}, a_{3}, \ldots be a sequence of positive integers where a1=∑i=0100ia_{1}=\sum_{i=0}^{100} i! and ai+ai+1a_{i}+a_{i+1} is an odd perfect square for all i≥1i \geq 1. Compute the smallest possible value of a1000a_{1000}.
Plain-text mathematical notation (without MathML)
Let a₁,a₂,a₃,… be a sequence of positive integers where a₁=∑_(i=0)¹⁰⁰i! and a_(i)+a_(i+1) is an odd perfect square for all i≥1. Compute the smallest possible value of a₁₀₀₀.
Original LaTeX notation
Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers where $a_{1}=\sum_{i=0}^{100} i$! and $a_{i}+a_{i+1}$ is an odd perfect square for all $i \geq 1$. Compute the smallest possible value of $a_{1000}$.

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Source and history

task_id
21f4c16b-d8d2-555f-a857-ece71db1074d
task_key
test--21f4c16b-d8d2-555f-a857-ece71db1074d
task_revision_id
1
config / split
/ test

Official source

initial import