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This question asks about sets such that every integer is expressible as a sum of four squares (of numbers from this set). Perhaps it would be possible to find a title which describes the question better than the current title Lagrange four squares theorem.
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Q: Lagrange four squares theorem

M. Farrokhi D. G.Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that is, every non-negative integer is a sum of squares of four elements of $X$. $\bullet$ Is $X=\{0,1...

 

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