I am not a mathematician, but while doing other work, I came across the Fortune conjecture. According to Wikipedia and other research, it seems that it has not yet been proven. I thought about the problem and within about 30 minutes came up with a partial solution that is so embarrassingly simple, I assume it must be incorrect. But, I am having trouble proving that my solution is incorrect, so I am asking for the community's help in explaining why I am wrong.
The conjecture, as I understand it, is that the difference between a primorial (Pn# where n is the number of primes in the primoral) and the next prime greater than the primorial will be prime.
My solution seems to work not just for the next prime greater than the primorial, but for all primes between the primorial and (the primorial + x^2) where x is the next prime number after nth prime. So, for example, if n=4, then the primorial is 2*3*5*7=210. The next prime after the nth prime is 11. 11^2=121. So, the difference between all the primes between 210 and (210+121=331) exclusive of 331 and the primorial 210+1 are primes. This can be checked quickly and easily on any spreadsheet and it does, in fact, hold true for n=1,2,3,4,5 (I did not go past that.)
The rationale for this is what is so simple. Using the previous example of n=4, the primorial is 210. The next prime is 211 (but Fortune himself excluded the primorial +1), so the next prime is 223. That is 210+13, so obviously the difference (13) is prime. In this specific scenario, all composite numbers 2 or greater up to 121 are divisible by one or more of the primes 2, 3, 5, or 7. How do we know this? Because it is well-established that any composite number has a factor that is equal to, or less than, the square root of the composite number. For example, the square root of 120 is 10.954 and the factors are 2-2-2-3-5.
Why does this matter? Because every number between 2 and 120 that is divisible by either 2, 3, 5, or 7, when added to 210 will produce a sum that is divisible by either 2, 3, 5, or 7. Why? Because 210 is divisible by each of those numbers. So, if you add a number that is also divisible by one of those numbers, the sum will also be divisible by the same number (for example, 210 = 7 x 30, so adding multiples of 7 to 210 like 217 = 7 x 31 or 224 = 7 x 32).
Therefore, for all numbers between 2 and 120, adding a composite to 210 will result in a composite, which is divisible by 2, 3, 5, or 7. As such, if there is a prime number between 212 and 330, it cannot be the result of adding a multiple of 2, 3, 5, or 7 to 210 or else the result would be a composite. Since every number that could be added to 210 with a result of 212-330 is either divisible by 2, 3, 5, or 7 or is prime, then only way to get a prime between 212-330 is to add a prime to 210.
As a result, it appears to me that I have proven that for every prime number between Pn# and Pn# + (n+1)^2 excluding Pn#+1 and Pn# + (n+1)^2 the number added to Pn# must be prime. This proof would apply to all n where there is even a single prime between Pn# and Pn# + (n+1)^2, not counting Pn#+1.
I guess we can determine whether there is a prime in the range I have set forth by looking at the prime gaps. The merit (Gn/ln(Pn)) where Gn is the prime gap and Pn is the prime, increases steadily with this series. Once you get to the tenth primorial (223092870), x^2 is 841. If there were not a prime in the next 841 numbers, then the merit of the previous prime would be greater than 841/ln(223092870) = 43.7. The merit increases as the primorial increases. Since the largest known merit is ~42, this essentially proves that there will be a prime in the first x^2 numbers after the primorial.