# Adding Primes to Produce a Prime

Inspiration for this problem comes from my good friend Johnny Aceti.

Consider numbers $x,y \in \mathcal{P}$ where $\mathcal{P}$ is the set of all prime numbers $\in \mathbb{N}$.

Conjecture 1. There exist consecutive $x,y \in \mathcal{P}$ such that $x+y \in \mathcal{P}$.

Is this true? Well lets see.

Every prime except 2 is odd. Let’s first look at adding together 2 odd primes. Odd numbers are equivalent to $1 \mod 2$, and if one adds together two numbers that are $1 \mod 2$, you get a sum that is $0 \mod 2$. This means the sum is even, and since we can’t add two primes to get 2, this means that you cannot add an even number of odd primes to produce a prime. This takes out just about every possibility for this conjecture to be true, but we haven’t considered 2 yet, since before we were only working with odd primes.

Let’s try the case where we use 2 as $x$.

$2 \equiv 0 \mod 2$ and every other odd prime is $1 \mod 2$. Therefore, if we added them, we could get a sum that is $1\mod 2$. So far so good, this means our sum is odd. But is it prime? The only instance involving 2 if $x, y$ have to be consecutive is $2+3=5$. 5 is prime, and that becomes the only $x,y$ that satisfy Conjecture 1.

But what if $x,y$ didn’t have to be consecutive? Does the use of 2 still work?

Well, the short answer is sometimes.

$\boldsymbol{2+3=5}$
$\boldsymbol{2+5=7}$
$2+7=9$

Uh oh. 9 isn’t prime. Therefore, the use of 2 as $x$ or $y$ only works sometimes. This problem isn’t very interesting with only two variables though. Let’s add more…

$***************************$

Consider consecutive numbers $x,y,z \in \mathcal{P}$ where $\mathcal{P}$ is the set  of all prime numbers $\in \mathbb{N}$.

$2+3+5=10$
$3+5+7=15$
$\boldsymbol{5+7+11=23}$
$\boldsymbol{7+11+13=31}$
$\boldsymbol{11+13+17=41}$
$13+17+19=49$
$\boldsymbol{17+19+23=59}$
$\vdots$

If we continue adding numbers in this way, how many times will the resulting sum be a prime? Is there any way to predict what triples of numbers produce primes and which do not?

Here are some more examples (all of which have prime sums):

$19 + 23 + 29 = 71$
$23 + 29 + 31 = 83$
$29 + 31 + 37 = 97$
$31 + 37 + 41 = 109$
$41 + 43 + 47 = 131$
$53 + 59 + 61 = 173$
$61 + 67 + 71 = 199$
$67 + 71 + 73 = 211$
$71 + 73 + 79 = 223$
$79 + 83 + 89 = 251$
$83 + 89 + 97 = 269$
$101 + 103 + 107 = 311$
$109 + 113 + 127 = 349$
$139 + 149 + 151 = 439$
$149 + 151 + 157 = 457$
$157 + 163 + 167 = 487$
$163 + 167 + 173 = 503$
$197 + 199 + 211 = 607$

Can we derive a pattern or a formula such that $x, y, z \in \mathcal{P}$ always holds?