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Factorial and Primes

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by gmatusa2010 » Fri Dec 03, 2010 9:36 pm
How many primes between 6!+2 and 6!+6. Is there an quick way to do this without multiply out 6!? Just wondering incase we get something much larger like 20!.
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by 4GMAT_Mumbai » Fri Dec 03, 2010 10:12 pm
Hi gmatusa2010 ... Interesting Q ... Thanks !

6! + 3 = 3 times (1.2.4.5.6 + 1) = 3 times <some integer>; Hence, not a prime

6! + 4 = 4 times (1.2.3.5.6 + 1) = 4 times <some integer>; Hence, not a prime

6! + 5 = 5 times (1.2.3.4.6 + 1) = 5 times <some integer>; Hence, not a prime

To make it more interesting, what would be the first number after 6! which is a prime number?

Would it be 6! + 7 ? Because 6! is not a multiple of 7? But, this only says that 6! + 7 is not a multiple of 7 ... It could be a multiple of some other integer ... 727 is indeed a prime # ... But, I do not suppose it can be generalized ... This is just a coincidence ...

In other words, is there a method to find the least value of b such that "a! + b" is a prime number ???
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by Rahul@gurome » Fri Dec 03, 2010 10:14 pm
gmatusa2010 wrote:How many primes between 6!+2 and 6!+6. Is there an quick way to do this without multiply out 6!? Just wondering in case we get something much larger like 20!.
(6! + 2) = (6*5*4*3*2*1 + 2) = 2*(6*5*4*3*1 + 1) => Multiple of 2
(6! + 3) = (6*5*4*3*2*1 + 3) = 3*(6*5*4*2*1 + 1) => Multiple of 3
(6! + 4) = (6*5*4*3*2*1 + 4) = 4*(6*5*3*2*1 + 1) => Multiple of 4
(6! + 5) = (6*5*4*3*2*1 + 5) = 5*(6*4*3*2*1 + 1) => Multiple of 5
(6! + 6) = (6*5*4*3*2*1 + 6) = 6*(5*4*3*2*1 + 1) => Multiple of 6

Thus there is no prime between (6! + 2) and (6! + 6).

Yes, there is short way.
In general, for any positive integer k, there is no prime between (k! + 2) and (k! + k). Proof is as above.
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by yellowho » Sun Feb 27, 2011 11:11 pm
Why K!+2 to K!+K?


Isn't it from K! to K! +K?


[quote="Rahul@gurome"][quote="gmatusa2010"]

Yes, there is short way.
In general, for any positive integer k, there is no prime between (k! + 2) and (k! + k). Proof is as above.[/quote]
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