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Omni-MATH / Call an ordered pair (a,b) of positive integers fantastic if and only if a,b≤10⁴ and gcd(a⋅n!−1,a⋅(n+1)!+b)>1 for infinitely many positive integers n. Find the sum of a+b across al…

Problem

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problem

Call an ordered pair (a,b)(a, b) of positive integers fantastic if and only if a,b≤104a, b \leq 10^{4} and gcd(a⋅n!−1,a⋅(n+1)!+b)>1\operatorname{gcd}(a \cdot n!-1, a \cdot(n+1)!+b)>1 for infinitely many positive integers nn. Find the sum of a+ba+b across all fantastic pairs (a,b)(a, b).
Plain-text mathematical notation (without MathML)
Call an ordered pair (a,b) of positive integers fantastic if and only if a,b≤10⁴ and gcd(a⋅n!−1,a⋅(n+1)!+b)>1 for infinitely many positive integers n. Find the sum of a+b across all fantastic pairs (a,b).
Original LaTeX notation
Call an ordered pair $(a, b)$ of positive integers fantastic if and only if $a, b \leq 10^{4}$ and $\operatorname{gcd}(a \cdot n!-1, a \cdot(n+1)!+b)>1$ for infinitely many positive integers $n$. Find the sum of $a+b$ across all fantastic pairs $(a, b)$.

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Source and history

task_id
5e6795e8-e096-54c3-a59b-37d3bce65625
task_key
test--5e6795e8-e096-54c3-a59b-37d3bce65625
task_revision_id
1
config / split
/ test

Official source

initial import