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Search for gaps between primes where each composite is divisible by increasing integers (2, 3, 4, ...), without using factorials

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Almost every text of number theory contains in its first chapters something similar to the following:

For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence $\,n!+2,\,n!+3,\,\ldots,\,n!+n\,$ the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of $\,n-1\,$ consecutive composite integers, and it must belong to a gap between primes having length at least $\,n$.

But what about gaps between primes where each composite is divisible by increasing integers $\,(2,\,3,\,4,\,\ldots)$, without using factorials?

Obviously, we shall say that $\,g_n\,$ is a "maximal gap", if $\,g_m\lt g_n\,$ for all $\,m\lt n$.

A list of the primes bounding the first maximal gaps (found searching into the first $\,10^7$ primes):

$(g=2)\;\;\;\;\;\;\,3,\,5\;\;\;\;\;\;\;\;(2|4)$

$(g=4)\;\;\;\;\;\;13,\,17\;\;\;\;\;\,(2|14,\,3|15,\,4|16)$

$(g=6)\;\;\;\;\;\;\,61,\,67\;\;\;\;\;(2|62,\,3|63,\,4|64,\,5|65,\,6|66)$

$(g=10)\;\;\;\;\;2521,\,2531\;\;\;\;\;(\ldots)$

$(g=12)\;\;\;\;\;471241,\,471253\;\;\;\;\;(\ldots)$

$(g=16)\;\;\;\;\;4324321,\,4324337\;\;\;\;\;(\ldots)$

$(g=18)\;\;\;\;\;110270161,\,110270179\;\;\;\;\;(\ldots)$

Here, instead, is a list of the frequencies ($f_g$) of all gaps (also not maximal) up to the first $\,10^7$ primes:

$(g=2)\;\;\;\;\;\;\,f_2=738597$

$(g=4)\;\;\;\;\;\;\,f_4=368781$

$(g=6)\;\;\;\;\;\;\,f_6=123052$

$(g=10)\;\;\;\;\;f_{10}=4136$

$(g=12)\;\;\;\;\;f_{12}=447$

$(g=16)\;\;\;\;\;f_{16}=17$

$(g=18)\;\;\;\;\;f_{18}=1$

Some remarks: starting from $g=6$, in order to have satisfied the divisibility by 5, the first prime must be congruent to 1 (mod 20); $\,g\,$ seems always long $\,p-1$ (or it is just a coincidence?); $\,f_2\sim 2f_4\sim 6f_6$; maybe, there are infinitely many maximal gaps.

Could the previous remarks (except for the first) be explained in any way?

Many thanks.

Almost every text of number theory contains in its first chapters something similar to the following:

For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence $\,n!+2,\,n!+3,\,\ldots,\,n!+n\,$ the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of $\,n-1\,$ consecutive composite integers, and it must belong to a gap between primes having length at least $\,n$.

But what about gaps between primes where each composite is divisible by increasing integers $\,(2,\,3,\,4,\,\ldots)$, without using factorials?

Obviously, we shall say that $\,g_n\,$ is a "maximal gap", if $\,g_m\lt g_n\,$ for all $\,m\lt n$.

A list of the primes bounding the first maximal gaps (found searching into the first $\,10^7$ primes):

$(g=2)\;\;\;\;\;\;\,3,\,5\;\;\;\;\;\;\;\;(2|4)$

$(g=4)\;\;\;\;\;\;13,\,17\;\;\;\;\;\,(2|14,\,3|15,\,4|16)$

$(g=6)\;\;\;\;\;\;\,61,\,67\;\;\;\;\;(2|62,\,3|63,\,4|64,\,5|65,\,6|66)$

$(g=10)\;\;\;\;\;2521,\,2531\;\;\;\;\;(\ldots)$

$(g=12)\;\;\;\;\;471241,\,471253\;\;\;\;\;(\ldots)$

$(g=16)\;\;\;\;\;4324321,\,4324337\;\;\;\;\;(\ldots)$

$(g=18)\;\;\;\;\;110270161,\,110270179\;\;\;\;\;(\ldots)$

Here, instead, is a list of the frequencies ($f_g$) of all gaps (also not maximal) up to the first $\,10^7$ primes:

$(g=2)\;\;\;\;\;\;\,f_2=738597$

$(g=4)\;\;\;\;\;\;\,f_4=368781$

$(g=6)\;\;\;\;\;\;\,f_6=123052$

$(g=10)\;\;\;\;\;f_{10}=4136$

$(g=12)\;\;\;\;\;f_{12}=447$

$(g=16)\;\;\;\;\;f_{16}=17$

$(g=18)\;\;\;\;\;f_{18}=1$

Some remarks: $\,g\,$ seems always long $\,p-1$ (or it is just a coincidence?); $\,f_2\sim 2f_4\sim 6f_6$; maybe, there are infinitely many maximal gaps.

Could the previous remarks be explained in any way?

Many thanks.

Almost every text of number theory contains in its first chapters something similar to the following:

For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence $\,n!+2,\,n!+3,\,\ldots,\,n!+n\,$ the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of $\,n-1\,$ consecutive composite integers, and it must belong to a gap between primes having length at least $\,n$.

But what about gaps between primes where each composite is divisible by increasing integers $\,(2,\,3,\,4,\,\ldots)$, without using factorials?

Obviously, we shall say that $\,g_n\,$ is a "maximal gap", if $\,g_m\lt g_n\,$ for all $\,m\lt n$.

A list of the primes bounding the first maximal gaps (found searching into the first $\,10^7$ primes):

$(g=2)\;\;\;\;\;\;\,3,\,5\;\;\;\;\;\;\;\;(2|4)$

$(g=4)\;\;\;\;\;\;13,\,17\;\;\;\;\;\,(2|14,\,3|15,\,4|16)$

$(g=6)\;\;\;\;\;\;\,61,\,67\;\;\;\;\;(2|62,\,3|63,\,4|64,\,5|65,\,6|66)$

$(g=10)\;\;\;\;\;2521,\,2531\;\;\;\;\;(\ldots)$

$(g=12)\;\;\;\;\;471241,\,471253\;\;\;\;\;(\ldots)$

$(g=16)\;\;\;\;\;4324321,\,4324337\;\;\;\;\;(\ldots)$

$(g=18)\;\;\;\;\;110270161,\,110270179\;\;\;\;\;(\ldots)$

Here, instead, is a list of the frequencies ($f_g$) of all gaps (also not maximal) up to the first $\,10^7$ primes:

$(g=2)\;\;\;\;\;\;\,f_2=738597$

$(g=4)\;\;\;\;\;\;\,f_4=368781$

$(g=6)\;\;\;\;\;\;\,f_6=123052$

$(g=10)\;\;\;\;\;f_{10}=4136$

$(g=12)\;\;\;\;\;f_{12}=447$

$(g=16)\;\;\;\;\;f_{16}=17$

$(g=18)\;\;\;\;\;f_{18}=1$

Some remarks: starting from $g=6$, in order to have satisfied the divisibility by 5, the first prime must be congruent to 1 (mod 20); $\,g\,$ seems always long $\,p-1$ (or it is just a coincidence?); $\,f_2\sim 2f_4\sim 6f_6$; maybe, there are infinitely many maximal gaps.

Could the previous remarks (except for the first) be explained in any way?

Many thanks.

Source Link

Search for gaps between primes where each composite is divisible by increasing integers (2, 3, 4, ...), without using factorials

Almost every text of number theory contains in its first chapters something similar to the following:

For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence $\,n!+2,\,n!+3,\,\ldots,\,n!+n\,$ the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of $\,n-1\,$ consecutive composite integers, and it must belong to a gap between primes having length at least $\,n$.

But what about gaps between primes where each composite is divisible by increasing integers $\,(2,\,3,\,4,\,\ldots)$, without using factorials?

Obviously, we shall say that $\,g_n\,$ is a "maximal gap", if $\,g_m\lt g_n\,$ for all $\,m\lt n$.

A list of the primes bounding the first maximal gaps (found searching into the first $\,10^7$ primes):

$(g=2)\;\;\;\;\;\;\,3,\,5\;\;\;\;\;\;\;\;(2|4)$

$(g=4)\;\;\;\;\;\;13,\,17\;\;\;\;\;\,(2|14,\,3|15,\,4|16)$

$(g=6)\;\;\;\;\;\;\,61,\,67\;\;\;\;\;(2|62,\,3|63,\,4|64,\,5|65,\,6|66)$

$(g=10)\;\;\;\;\;2521,\,2531\;\;\;\;\;(\ldots)$

$(g=12)\;\;\;\;\;471241,\,471253\;\;\;\;\;(\ldots)$

$(g=16)\;\;\;\;\;4324321,\,4324337\;\;\;\;\;(\ldots)$

$(g=18)\;\;\;\;\;110270161,\,110270179\;\;\;\;\;(\ldots)$

Here, instead, is a list of the frequencies ($f_g$) of all gaps (also not maximal) up to the first $\,10^7$ primes:

$(g=2)\;\;\;\;\;\;\,f_2=738597$

$(g=4)\;\;\;\;\;\;\,f_4=368781$

$(g=6)\;\;\;\;\;\;\,f_6=123052$

$(g=10)\;\;\;\;\;f_{10}=4136$

$(g=12)\;\;\;\;\;f_{12}=447$

$(g=16)\;\;\;\;\;f_{16}=17$

$(g=18)\;\;\;\;\;f_{18}=1$

Some remarks: $\,g\,$ seems always long $\,p-1$ (or it is just a coincidence?); $\,f_2\sim 2f_4\sim 6f_6$; maybe, there are infinitely many maximal gaps.

Could the previous remarks be explained in any way?

Many thanks.