I have been working on a conjecture the past couple days and would like assistance in determining whether or not it is true or not.
The question is as follows. Given a distinct set of ascending prime numbers from 2 to some $P_n$ (where $P_n$ is the nth prime) determine the length of the longest theoretical string of consecutive adjacent composite numbers that are dependent on that set. This means a string of consecutive composite numbers such that each composite in the string is a multiple of at least 1 number in your prime factor set. These composite numbers do NOT have to exclusively share factors in this set, only 1 shared factor is necessary. This means that if the set is {2, 3} then the composites dependent on this set are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ... and the first consecutive adjacent composite string we could view is 8, 9, 10 which is of length 3. Additionally, we can see that the first composite not dependent on the set {2,3} is actually 25, which is the square of the next largest prime.
An example:
The hypothesis that I have come up with is that the length must exist within the range of $2(P_{n-1}) - 1$ and $2(P_n - 2) - 1$. First, this means that the length of the greatest string can be calculated exactly so long as the largest 2 primes in your factor set have a prime gap of 2. Second, the exact relationship of the length of your string, represented as x, to these equations is as follows. $2(P_{n-1}) - 1 ≤ x < 2(P_n – 2) - 1$
Using a program, I was able to confirm that the largest observable string under these conditions for all sets up to the set containing 29, fall perfectly within this range, most being the lowest value in this range equal to $2(P{n-1}) - 1$ where $P_{n-1}$ is the penultimate prime in the set. They are listed below:
{2} - Largest string is of length 1
{2, 3} - Largest string is of length 3
{2, 3, 5} - Largest string is of length 5
{2, 3, 5, 7} - Largest string is of length 9
{2, 3, 5, 7, 11} - Largest string is of length 13
{2, 3, 5, 7, 11, 13} - Largest string is of length 21
{2, 3, 5, 7, 11, 13, 17} - Largest string is of length 25
{2, 3, 5, 7, 11, 13, 17, 19} - Largest string is of length 33
{2, 3, 5, 7, 11, 13, 17, 19, 23} - Largest string is of length 39
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29} - Largest string is of length 45
These observable lengths were found by iterating over all numbers from 1 to the nth primorial and keeping track of the largest string observed. For example, the first acceptible string of max length found by the program for the set {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} lies between 417086648 and 417086692 and has a length of 45.
Moreso, I have determined a way to find the most optimal string of composites for any set where the largest primes have a prime gap of 2. The method is as follows. Create a number line and start from the very center. This value will be referred to as your "zero". List the prime factors of your 0 as every factor in your factor set excluding the largest 2 primes. Next, place your "ones". These numbers will be to the immediate left and right of your "zero" on the number line, and they will be the 2 largest primes in your set. Next, simply begin counting outward from 2 to the number one less than your lowest "one". Refer to the image shared previously for the set {2, 3, 5, 7, 11}. Note the center is your "zero" which has factors 2, 3, and 5. To the left and right are the "one"s 11 and 7. Then, you count outward from there, 2, 3, 4, 5, 6 and place the prime factors of each of these numbers on the number line in the respective position. This method should always give you the longest possible string of consecutive composite values that are multiples of any value in your set so long as the prime gap between the largest two primes is 2. If the prime gap is larger than 2, then the length x will be in a range determined by $2(P_{n-1}) - 1 ≤ x < 2(P_n – 2) - 1$. This is because the optimal pattern relies on the smaller "one" value to be exactly 2 below the other, and prime. You could model the same pattern such that your "one"s are instead $P_n$ and $P_n - 2$ where $P_n - 2$ is NOT prime and see that it would not logically be possible since that would conflict with the distribution of all prime factors already established by the center point or "zero" of the pattern.
The hypothesis shared is predicated on the idea that the pattern exhibited in the above paragraph will always be the most optimal pattern you could create using the given set. Disproving this hypothesis would be as simple as finding a theoretical series of multiples dependent on the set $\{2 - P_n\}$ that could all exist on a real number line which has a length exceeding $2(P_n – 2) - 1$.
I really need assistance proving this hypothesis. I can not think of any organization of multiples you could ever come up with that will exceed this value with the restrictions I have given. If someone could either prove or disprove this hypothesis, that would be great.
Also, assuming this could be proven, I believe that this conjecture would have implications on the upper bounds of prime gaps up to a given number. Specifically, we can show that for any square of a prime $P_n$, all numbers below that must either be the unit 1, a prime number, or a composite dependent on the set of all primes less than $P_n$ similar to the limitations of the conjecture above, and our conjecture would supply the length of the longest possible prime gap that could be made under those limitations. This may also act as a proof of Legendre's conjecture since it would show that the largest prime gap possible between $n^2$ and $(n + 1)^2$ is equal to $2(P - 2)$ where P is the closest prime less than n, and is hence, smaller than the gap between $n^2$ and $(n + 1)^2$ which is itself $2n + 1$.