My Math Forum Primorial + prime = prime

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 September 23rd, 2008, 08:46 AM #1 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 Primorial + prime = prime Hi, If you add to primorial p! (for p=1 to eight) the prime 881 you will find 8 prime numbers in a row. p! ..... p!+881 2........... 883 6........... 887 30..........911 210.........1091 2310........3191 30030......30911 510510.....511391 9699690...9700571 Is there better number to find more in row?
 September 23rd, 2008, 09:25 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Primorial + prime = prime The product of the primes up to p, inclusive, is denoted p# and called the primorial. I see that 41 has the same property you ascribe to 881. 2# + 3 2# + 5 3# + 5 2# + 11 3# + 11 5# + 11 2# + 17 3# + 17 5# + 17 7# + 17 2# + 41 3# + 41 5# + 41 7# + 41 11# + 41 13# + 41 17# + 41 19# + 41 2# + 86351 3# + 86351 5# + 86351 7# + 86351 11# + 86351 13# + 86351 17# + 86351 19# + 86351 23# + 86351 29# + 86351 2# + 235313357 3# + 235313357 5# + 235313357 7# + 235313357 11# + 235313357 13# + 235313357 17# + 235313357 19# + 235313357 23# + 235313357 29# + 235313357 31# + 235313357 To check a number: Code: try(n)=my(pr=2,p=2);while(isprime(pr+n),p=nextprime(p+1);pr*=p);primepi(p)-1 To look for records: Code: rec=0;forprime(p=2,2.5e8,k=try(p);if(k>rec,rec=k;print(k" "p))) If you wanted to check for primorial + n = prime, rather than only consider adding primes, replace "forprime" in the above with "for".
 September 23rd, 2008, 09:41 AM #3 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 Re: Primorial + prime = prime I have tried in the past a lot of combinations such as 2^n + p (n varying from 1 to k) and others. I had never found more than 12 in a row. I have posted my example to discuss why there is a finite limit.
 September 23rd, 2008, 01:06 PM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Primorial + prime = prime Why do you think there's a limit? I see no reason there would be one. The whole range I searched was productive. Sure, the primes thin out, but that just means you have to search harder.
 June 3rd, 2017, 03:29 PM #5 Banned Camp   Joined: Dec 2013 Posts: 1,117 Thanks: 41 Here is another idea of mine requiring deep study Good luck!

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