Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.

Question

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.

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I cannot give a proof to this, although I have try on some small examples.

I also notice that $\pi(x)\sim x/\log x$ and there are approximately $\sqrt x$ perfect square smaller than $x$. I have a feel that because $\sqrt x=\mathcal O(x/\log x)$, there will be more perfect squares, two or three, between $s_n$ and $s_{n+1}$ when $n$ gets large.

Any suggestion?

Answer 1

After some discussion with my professor, we have solved this problem by using some elementary arguments.

We first prove the following lemma.

Lemma : For $n\geq 4$ we have $$s_n<\bigg(\frac{p_{n+1}-1}{2}\bigg)^2.$$ To show this we first see $s_4=2+3+5+7=17$ and $(p_{5}-1)/2=5$, so obviously $17< 5^2$. Now if there is $m\geq 4$ such that $s_m<[(p_{m+1}-1)/2]^2$, then $$s_{m+1}=s_m+p_{m+1}<\bigg(\frac{p_{m+1}-1}2\bigg)^2+p_{m+1}=\bigg(\frac{p_{m+1}+1}2\bigg)^2\leq \bigg(\frac{p_{m+2}-1}{2}\bigg)^2$$ since for primes $\geq 11$, every pair of consecutive primes has least gap equals to $2$, the lemma's proof is completed now.

Now choose $n\geq 4$, let $a$ be the largest integer satisfy $a^2\leq s_n$, then by definition we have $(a+1)^2> s_n$. Now we have $$a^2\leq s_n< \bigg(\frac{p_{n+1}-1}2\bigg)^2\implies a< \frac{p_{n+1}-1}{2}$$ therefore $2a+1< p_{n+1}$. Combining these we have $$(a+1)^2=a^2+2a+1< s_n+p_{n+1}=s_{n+1}$$ So we have proved that $s_n< (a+1)^2< s_{n+1}$.

We have $s_1=2, s_2=5, s_3=10, s_4=17$, so $$s_1< 2^2< s_2< 3^2< s_3< 4^2< s_4$$ completely proved this statement.

Answer 2

The sum of the first $n$ primes is

$$ s_n = \frac{n^2}{2}\Big(\log n + \log\log n - \frac32 + \frac{\log\log n}{\log n} - \frac{5}{2\log n} + o(1)\Big) $$

Which gives $$ s_{n+1} - s_n \approx n^2 (\log n + \log\log n - 3/2) > n^2 $$

for all $n > 10$. For $n = 1,2,3$ we can verify it manually. Since the gap between $s_n$ and $s_{n+1}$ jumps by more than $n^2$ the interval $s_n, s_{n+1}$ will contains a square. More specifically we can prove that

$$ s_n < (n+1)^2 < s_{n+1} $$