Prime Number Sum Sequence (Amateur)

Question

SOLVED: This is false

Beginning with 3, add the next consecutive prime (2) and then take that sum (5) and add that to next consecutive prime (3) to get (8), and so on...

$$ 3 + 2 = 5 $$ $$ 5 + 3 = 8 $$ $$ 8 + 5 = 13 $$ $$ 13 + 7 = 20 $$ $$ 20 + 11 = 31 $$ $$ 31 + 13 = 44 $$ $$ 44 + 17 = 61 $$ $$ 61 + 19 = 80 $$ $$ 80 + 23 = 103 $$

Could you prove that every odd number in this sequence is prime AND that every even number (except 2) minus 1 is prime as well? i.e (8-1=7)(20-1=19)

Answer 1

I will assume the suggested sequence is:

Starting with $5$, add primes starting from $3$; equivalently, starting from $3$, add primes starting from $2$, the first one.

However, the $21$st partial sum of primes is $712$, and $712+3=715$ which is obviously divisible by $5$.

Answer 2

I love elementary conjectures, and I always try to encourage students in my elementary number theory classes to conjecture away, right or wrong. Here, it happens that both of your conjectures are wrong. The first several primes are

$$2,3,5,7,11,13,17,19,23,$$

with partial sums

$$2,5,10,17,28,41,58,\color{#aa0000}{77}, 100.$$

I've highlighted $77$ in $\color{#aa0000}{\text{red}}$ because it's not prime. Also, every even number except for $2$ on this list fails your second conjecture, as $9, 27, 57,$ and $99$ are all composite.