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?

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