# Infinite set of positive integers such that no subset sum is a square

Find an infinite set of positive integers $$S$$ such that them sum of the elements of any finite subset of $$S$$ is not a perfect square.

I've had one person tell me that $$\{3\cdot 4^k : k\in \mathbb{N}\}$$ works, but I don't see why. What are some other sets that work? Why does this one work?

• $\{2^{2k+1} : k\in \mathbb{N}\}$ would be another simple example – Sil Mar 31 at 16:04

Let $$a^2=\sum_{i=1}^n3*4^{k_i}$$ with $$k_i, then dividing by $$4^{k_1}$$ we get ( setting $$\frac{a}{2^{k_1}}=u$$ ): $$u^2=\sum_{i=1}^n3*4^{k_i-k_1}=3+\sum_{i=2}^n3*4^{k_i-k_1}$$, and thus $$u^2=3\mod{}4$$ which is a contradiction since $$u^2=0$$ or $$1\mod{}4$$.
In general you can set $$A=\{ab^i,i\in\mathbb{Z}\}$$ where $$a$$ is not a quadratic residue $$\mod{}b$$ or equivalently $$a\ne{}x^2\mod{}b$$ for every $$x\in{}\mathbb{Z/}b\mathbb{Z}$$.