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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