In this post I want to present the following conjecture related to odd-indexed prime numbers (see OEIS sequence A031368): All the odd-indexed prime numbers larger than 2 can be obtained from all the previous prime numbers using addition and subtraction.
The conjecture was inspired by Curiosa 67 from Scripta Mathematica ( page 159, Volume 7, 1940). I wrote 2 posts (part 1 and part 2) that discussed the sums of prime numbers mentioned in Curiosa 67.
Examples
The numbers in the summations are ordered in increasing order (according to their absolute value):
p(3)=5=2+3
p(5)=11=2-3+5+7
p(7)=17=2+3-5-7+11+13
p(9)=23=2-3+5-7+11+13-17+19
p(11)=31=2+3+5-7+11-13+17+19+23-29
p(13)=41=2-3+5-7+11+13+17+19+23+29-31-37
p(15)=47=-2+3-5-7+11+13+17+19+23-29-31+37+41-43
Final Notes
Another question to consider: Are the summations always unique?
I am not sure if this conjecture was discussed or proved in the existing literature. Nonetheless, I believe that these types of sums deserve some attention. Maybe they will lead to new interesting conjectures.
Addendum: OEIS Sequence A022894
I started a Reddit discussion on this topic. An OEIS veteran editor mentioned that the OEIS sequence A022894 tackled this conjecture. The sequence is about the number of solutions, and we can see that only 5,11 have unique solutions. 17 can be obtained 2 ways from the previous prime numbers. The number of solutions increase rapidly for higher prime numbers.