# Are there infinitely many additive prime numbers? [closed]

Notation: For any $n\geq 1$ let $p_{n}$ be $n$-th prime number.

Definition: A prime number $p_{n}$ is "additive" iff $p_{n}=\sum_{i<n} p_{i}$

Example: $5$ is an additive prime number.

Question (1): What is the least additive prime number larger than $5$?

Question (2): Are there infinitely many additive prime numbers?

Question (3): If $p_{n}$ be an additive prime number, is $n$ a prime number too?

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## closed as off-topic by Joseph Van Name, Andres Caicedo, Igor Pak, David White, Qiaochu YuanOct 28 '13 at 2:26

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Joseph Van Name, Andres Caicedo, Igor Pak, David White, Qiaochu Yuan
If this question can be reworded to fit the rules in the help center, please edit the question.

This is more appropriate for Math StackExchange (i.e., it is not a research-level question, but rather seems to be something at the level of idle curiosity, though I have to confess that the motivation for wondering about it escapes me). –  Marguax Oct 27 '13 at 23:14
Perhaps it's natural that when one asks five questions per day, they won't all be great. –  Michael Zieve Oct 27 '13 at 23:50
How did this question manage to receive 4 upvotes? –  Joseph Van Name Oct 28 '13 at 13:15
@JosephVanName: I think its really interesting. I introduced a very simple and natural property for a prime number which determines the number 5 "uniquely". It is rather unusual because in this kind of questions in number theory one can find no solution or infinitely many. Note that the answer is simple but not immediate, trivial or natural for researchers who are not professional number theorist. –  Ali Sadegh Daghighi Oct 28 '13 at 15:34
For future reference, let me suggest how you can try to answer such questions on your own. Since your question is about prime numbers, the first thing to do is to see whether basic facts about prime numbers provide an approach. There are many good references, for instance the wikipedia article on the topic, or anything you find via searching the web. Any such reference will point you to the Prime Number Theorem, which answers your question. It is better to try to find an answer on your own before posing a question to the world. –  Michael Zieve Oct 28 '13 at 22:49

As $p_n$ is asymptotically $n\log n$, the inequality $p_n\geq p_{n-1}+p_{n-2}$ only has finitely many solutions. In particular, there are only finitely many additive prime numbers, and these are easy to find by more precise (standard) estimates for $p_n$.
Added. Here are more details. By the classical paper of Rosser-Schoenfeld (Illinois J. Math. 6 (1962), 64-94.) we have for $n\geq 6$ the bound $$n\log n < p_n < n(\log n+\log\log n).$$ From here it is easy to see that that the inequality $p_n\geq p_{n-1}+p_{n-2}$ implies $n\leq 7$, hence there are no additive prime numbers beyond the seventh prime, which is $17$. From here it follows by a manual check that $5$ is the only additive prime.
@MarianoSuárez-Alvarez: I wrote a simple program which checked the numbers less than $10^{6}$ but didn't find anything. –  Ali Sadegh Daghighi Oct 27 '13 at 23:27