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Statement

Denote $p_n$ the $n$th prime. Consider the set $$\mathcal{P}=\{\sum_{i=1}^np_i:n\in\mathbb{N}\}$$ Prove that $|\mathcal{P}\cap\mathbb{P}|=\infty$

Why am I studying this

First of all, I find it an itriguing and rather unusual result. Moreover, this might provide us with a formula for infinitely many primes (such formula is not known except for the $ak+b$ from Dirichlet's theorem). I also want to see how we can generate primes by adding numbers, so this was a natural idea. To conclude, I also highly highly suspect these primes have some extra properties from normal ones. For now observations and proofs should lead the study.

Computational arguments

First of all, here are the first $200$ values for $\sum_{i=1}^{n}p_i$:

5 10 17 28 41 58 77 100 129 160 197 238 281 328 381 440 501 568 639 712 791 874 963 1060 1161
 1264 1371 1480 1593 1720 1851 1988 2127 2276 2427 2584 2747 2914 3087 3266 3447 3638 3831 4028 
4227 4438 4661 4888 5117 5350 5589 5830 6081 6338 6601 6870 7141 7418 7699 7982 8275 8582 8893 
9206 9523 9854 10191 10538 10887 11240 11599 11966 12339 12718 13101 13490 13887 14288 14697 
15116 15537 15968 16401 16840 17283 17732 18189 18650 19113 19580 20059 20546 21037 21536 22039 
22548 23069 23592 24133 24680 25237 25800 26369 26940 27517 28104 28697 29296 29897 30504 31117 
31734 32353 32984 33625 34268 34915 35568 36227 36888 37561 38238 38921 39612 40313 41022 41741 
42468 43201 43940 44683 45434 46191 46952 47721 48494 49281 50078 50887 51698 52519 53342 54169 
54998 55837 56690 57547 58406 59269 60146 61027 61910 62797 63704 64615 65534 66463 67400 68341 
69288 70241 71208 72179 73156 74139 75130 76127 77136 78149 79168 80189 81220 82253 83292 84341 
85392 86453 87516 88585 89672 90763 91856 92953 94056 95165 96282 97405 98534 99685 100838 
102001 103172 104353 105540 106733 107934 109147 110364

These are the primes that were generated:

    2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, 25237, 28697, 32353, 37561, 38921, 43201, 44683, 55837, 61027, 66463, 70241, 86453, 102001, (and some others: 109147, 116533, 119069, 121631, 129419, 132059, 263171, 287137, 325019, 329401, 333821, 338279, 342761)

Note that because I lack C++ and Python knowledge, i had to check for primes manually/using a website so it is a hard job. It seems that these primes are scarce, but even with small computational evidence, i think there are infinitely many. Any more of these primes posted in the comments will highly help.

Generalizations

Of course, there are many ways to generalize this. I will state 3, which might be of interest. However, for the sake of simplicity, i will only leave the first statement as a conjecture. These new results are quite similar so I think a hypothetical proof would not differ too much.

Generalization 1:

The same question as the original one, but this time $$\mathcal{P_k}=\{\sum_{i=k}^np_i:n\in\mathbb{N}\}$$ Prove that $\forall k$, $|\mathcal{P_k}\cap\mathbb{P}|=\infty$

Generalization 2:

The same question as the original one, but this time $$\mathcal{P^l}=\{\sum_{i=1}^np_i^l:n\in\mathbb{N}\}$$ Prove that $\forall k$, $|\mathcal{P^l}\cap\mathbb{P}|=\infty$

Generalization 3:

A combination of generalization 1 generalization 2 $$\mathcal{P_k^l}=\{\sum_{i=k}^np_i^l:n\in\mathbb{N}\}$$ Prove that $\forall k$, $|\mathcal{P_k^l}\cap\mathbb{P}|=\infty$

I am not providing computational arguments for these ones.

Further research on the original hypothesis

Let us call a prime $p\in\mathcal{P}$ an exquisite prime. Here i will formulate properties of exquisite primes. (Currently N/A)

Observations

Currently we do not have any hypothesis/observation on the behavior/the primes. Again, please leave anything you noticed in the comments.

This will be frequently be updated as progress is made.

More Resources

https://arxiv.org/abs/1804.04198

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  • 1
    $\begingroup$ See also this paper and the OEIS sequence A013918. $\endgroup$ – Peter Foreman Jun 17 at 17:25
  • $\begingroup$ Well, it doesnt, as in that post the question is too, unanswered $\endgroup$ – Vlad Jun 17 at 17:40
  • $\begingroup$ It's a duplicate ... nobody can answer that question at the moment! MSE is online for more than 10 years, the likelihood of asking a duplicate is increasing each year. It is always advisable to search before asking. $\endgroup$ – rtybase Jun 17 at 17:43
  • $\begingroup$ Here is another one ... $\endgroup$ – rtybase Jun 17 at 17:48
  • $\begingroup$ This is one of the statements about primes that are "almost surely true", but a proof is "completely out of reach" or like Erdoes would say it : "Mathematics is not ready yet for such problems". It is not difficult to program a routine in PARI/GP searching for such primes. $\endgroup$ – Peter Jun 19 at 11:37