Disclaimer: I’m an amateur, and have no advanced knowledge of math, so please forgive my ignorance as I’m just curious to know if I’ve stumbled upon something or not.
Prime Numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109
Applying the prime numbers to the list of prime numbers yields the following when they are aligned:
3, 5, 11, 17, 31, 41, 59, 67, 83, 109
- 3 is the 2nd prime
- 5 is the 3rd prime
- 11 is the 5th prime
- 17 is the 7th prime
- 31 is the 11th prime
- 41 is the 13th prime
- 59 is the 17th prime
- 67 is the 19th prime
- 83 is the 23rd prime
- 109 is the 29th prime
So three is the first in our new list because it is the 2nd prime and 2 is our first prime number.
Then by adding the original list of primes up until each of the new primes in our list we are left with this:
2 + 3 = 5
2 + 3 + 5 = 10
2 + 3 + 5 + 7 + 11 = 28
2 + 3 + 5 + 7 + 11 + 13 + 17 = 58
2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 = 160
Next five sums are: 238, 440, 568, 696, and 1002
Question: Will this pattern always produce an even sum?
Notice you keep adding two primes at a time. All primes, except for $2$, are odd. The sum of two odd numbers is even. Then you add the first prime, $2$, which is even, meaning the sum is still even. So you will always end up with an even sum.
P.S. Keep up with your investigations! The deepest understanding of Mathematics comes from playing around with things yourself and looking for patterns, asking your own questions, and exploring the subject in your own way.