# Adding and Subtracting the First n Prime Numbers to Get the Next Prime Number

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.