Must Have All Integers

Number Theory Level 5

Given two integers \(m, n,\) a set \(\mathbb{S}_{m,n}\) consisting of integers is defined as follows:

  • \(m, n \in \mathbb{S}_{m,n}\)
  • Consider an integer \(x \neq m,n.\) Then, \(x \in \mathbb{S}_{m,n}\) if and only if there exist two \(a, b \in \mathbb{S}\) and a \(k \in \mathbb{Z}\) such that \(a^2+kab+b^2=x.\)

Find the number of positive integer pairs \((m,n)\) such that \(1 \leq m < n \leq 5\) and for all possibilities of \(\mathbb{S}_{m,n},\) \(\mathbb{S}_{m,n} = \mathbb{Z}.\)

Details and assumptions

  • The condition in the last sentence implies that \(\mathbb{S}_{m,n}\) must contain all integers.
  • \(a\) and \(b\) need not be distinct in the second condition.
  • This problem is adapted from ISL.

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