Finding a special prime

Question

The prime number $$p=82\ 954\ 517$$ has the property that the numbers $$2!+p,3!+p,\cdots , 11!+p$$ are all prime, but $12!+p$ is composite. Upto $10^{10}$, the only other prime with this property is $105\ 204\ 557$

Does a prime $p$ exist such that $$2!+p,3!+p,\cdots , 12!+p$$ are all prime ? If yes, which is the smallest such prime ? Such a prime must exceed $10^{10}$

Update : The prime $$p=79\ 017\ 245\ 897$$ is even better than what I wanted. $j!+p$ is prime for $j=2,3,4,\cdots,13$. Now it remains to find the minimum primes for the limit $12$ and the limit $13$

Answer 1

I've only found the number you've found yourself as well:

79017245897 is a candidate
    79017245897 + 2! (== 2) = 79017245899 is prime
    79017245897 + 3! (== 6) = 79017245903 is prime
    79017245897 + 4! (== 24) = 79017245921 is prime
    79017245897 + 5! (== 120) = 79017246017 is prime
    79017245897 + 6! (== 720) = 79017246617 is prime
    79017245897 + 7! (== 5040) = 79017250937 is prime
    79017245897 + 8! (== 40320) = 79017286217 is prime
    79017245897 + 9! (== 362880) = 79017608777 is prime
    79017245897 + 10! (== 3628800) = 79020874697 is prime
    79017245897 + 11! (== 39916800) = 79057162697 is prime
    79017245897 + 12! (== 479001600) = 79496247497 is prime
    79017245897 + 13! (== 6227020800) = 85244266697 is prime

I did this using an exhaustive search (I've calculated all the prime numbers up to 160,000,000,000 and then checked for each prime if the above sums are in the list). This also shows that 79,017,245,897 is the smallest prime with this property.

I'll repeat it now for factorials up to 12! and add it here as soon as it has finished. Will there be a maximum? I mean will there be a number $n$ such that there doesn't exist a prime $p$ for which $p + i!$ is prime for all $i \in \{2, \ldots, n\}$.

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As promised, my results for $n = 12$:

 79017245897 is a candidate
148165314371 is a candidate

IOW, 79,017,245,897 is the smallest both for $n = 12$ and $n = 13$.

Answer 2

Just a few restrictions:

• first leads to p is 5 mod 6
• second eliminates 29 mod 30
• third eliminates 11 mod 30
• fourth eliminates 6 mod 7 ( aka 167,83 mod 210)
• fifth eliminates 1 mod 7 ( aka 197, 113 mod 210)
• sixth eliminates 9 mod 11
• seventh eliminates 6 mod 11
• eighth eliminates 10 mod 11
• ninth eliminates 1 mod 11
• tenth eliminates 12 mod 13
• eleventh eliminates 1 mod 13
• twelvth eliminates 14 mod 17
• thirteenth eliminates 9 mod 17
• fourteenth eliminates 16 mod 17
• fifteenth eliminates 1 mod 17
• sixteenth eliminates 18 mod 19
• seventeenth eliminates 1 mod 19
• eighteenth eliminates 19 mod 23
• nineteenth eliminates 12 mod 23
• twentieth eliminates 22 mod 23
• twenty first eliminates 1 mod 23
• twenty second eliminates 7 mod 29
• twenty third eliminates 23 mod 29
• twenty fourth eliminates 24 mod 29
• twenty fifth eliminates 15 mod 29
• twenty sixth eliminates 28 mod 29
• twenty seventh eliminates 1 mod 29
• twenty eighth eliminates 30 mod 31
• twenty ninth eliminates 1 mod 31
• thirtieth eliminates 4 mod 37

okay I have messed up ( prior) we need -n! mod q# eliminated. table updated (fixed) and extended.