Inspiration for this problem comes from my good friend Johnny Aceti.
Consider numbers where is the set of all prime numbers .
Conjecture 1. There exist consecutive such that .
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 , and if one adds together two numbers that are , you get a sum that is . 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 .
and every other odd prime is . Therefore, if we added them, we could get a sum that is . So far so good, this means our sum is odd. But is it prime? The only instance involving 2 if have to be consecutive is . 5 is prime, and that becomes the only that satisfy Conjecture 1.
But what if didn’t have to be consecutive? Does the use of 2 still work?
Well, the short answer is sometimes.
Uh oh. 9 isn’t prime. Therefore, the use of 2 as or only works sometimes. This problem isn’t very interesting with only two variables though. Let’s add more…
Consider consecutive numbers where is the set of all prime numbers .
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):
Can we derive a pattern or a formula such that always holds?