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September 23rd, 2008, 09:46 AM   #1
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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?
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September 23rd, 2008, 10:25 AM   #2
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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".
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September 23rd, 2008, 10:41 AM   #3
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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.
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September 23rd, 2008, 02:06 PM   #4
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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.
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June 3rd, 2017, 04:29 PM   #5
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Here is another idea of mine requiring deep study
Good luck!
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