MathDB

For each positive integer

k

, let

n_k

be the smallest positive integer such that there exists a finite set

A

of integers satisfy the following properties:
[*]For every

a\in A

, there exists

x,y\in A

(not necessary distinct) that

n_k\mid a-x-y

[/*] [*]There's no subset

B

A

that

|B|\leq k

and

n_k\mid \sum_{b\in B}{b}.

Show that for all positive integers

k\geq 3

, we've

n_k<\Big( \frac{13}{8}\Big)^{k+2}.

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