# Sum of all subsets of the set of even coprime integers relative to a power of a prime

by Ale Tolcachier   Last Updated June 30, 2020 02:20 AM

Let $$p$$ be an odd prime and $$k\in \mathbb{N}$$.

Let $$S$$ be the set of even coprime integers relative to $$p^k$$.

Question: If we take all the sums of all subsets of $$S$$ (with at least one element and without repeating elements), can we get all the even numbers up to the sum of all elements of $$S$$?

I could prove it easily by induction in the case of $$k=1$$ and I checked it for some low powers for example $$9,25,27$$. I think that this is true but I don't know how to manage a proof for $$k\geq 2$$ or find a counterexample. Maybe this is related to the Euler's totient function $$\varphi$$.

Any comment or sugerence will be appreciated. Thanks!

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