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Can we find, using elementary ways, an additive basis of order $2$, $(u_n)_{n\geqslant1}$, such that $\lim\limits_{n\rightarrow+\infty}(u_{n+1}-u_n)=+\infty$ ?

If $\alpha\in\left]1,\frac32\right[$, the sequence $\left(\lfloor n^{\alpha}\rfloor\right)$ works (have a look here), but the proof is a little complicated for me...

  • I don't think it even exists. – Crostul Jul 10 '18 at 17:14
  • I am not sure what you want in here: Is it about "Given $u_n$ with the condition, any natural number $m$ is expressed as a sum of $u_k+u_l$?" – Sungjin Kim Jul 13 '18 at 00:01
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    Also, I'd like to know why you think your question (I) does not answer (II). – Sungjin Kim Jul 13 '18 at 00:04
  • Yes, my question was not clear. I fixed it. –  Jul 13 '18 at 06:11
  • Similar to Goldbach's conjecture would be something like if $\ \left{\left\lfloor \frac{1}{2} n\log n \right \rfloor \right}\ $ is an additive basis of order two. I don't know what the status of such a conjecture is. – Adam Rubinson Dec 29 '22 at 23:02

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I finally found this article : Cassels bases (cf. Theorem 6, p. 11).