A Math... Question :)

[Xeda112358]

I have a little math challenge for y'all… This one isn't difficult so long as you don't overthink it, but if you do (like me) you will probably laugh at the simple solution. My hope is that you are like me and this keeps you occupied for a little while :

Find 132 consecutive composite natural numbers.

• Natural numbers are {1,2,3,4,…}
• A composite number is a number that isn't prime like 15 (15=3*5)
• An example of 12 consecutive composites is: {114,115,116,117,…,125,126}

I have two solutions that I came up with (one of them required Latex)
Have fun!?

[sqrt(Time)]

I got an answer almost instantly, but that's just because I've seen this exact kind of problem done before. ^_^ (I guess that's cheating)
It is indeed fun...
I'm wondering what your second solution is?

[boot2490]

Here:
-132 through 0
Its too easy

P.S. I did overthink it. I looked of a list with 50 million prime numbers and couldn't find such a gap.

EDIT: Oh wait... those aren't natural...

[Xeda112358]

Is that a negative 132?

• Natural numbers are {1,2,3,4,…}

Also, such a gap does indeed exist In fact, there exists much more massive gaps

EDIT: Also, sqrt(Time), if you want to pm me your solution, feel free or you can wait until others have found it

[Ashbad]

Hmm, I'll be interested to see a solution.  I've never seen this problem before, so I don't know the solution or a way to obtain it; hence, I've been playing around with some ideas in GHCi, to no current avail.

[boot2490]

Yeah,
_m39 through that plus 132.
Primes.zip has a text file in it with all of the digits. It is over 4 megabytes.
Whoa.

[Xeda112358]

is that a mersenne prime (out of curiosity)?

[boot2490]

Si senor.

[Xeda112358]

Okay, well I do not have the computational power to test those values, but if you can show me that the next prime is greater than 133 digits away, I will accept that Note that if you are comparing, say, m34 to m35, that does not say that the next prime after m34 is m35 It just says that is the next mersenne prime

[boot2490]

m39 is THE LARGEST prime number known to man. If the next one was less than 133 away, we would have found it by now.
Also, What is your answer?

P.S. How is my sig? suggestions?

[Xeda112358]

Actually, that is not true That might be the largest known prime (I think there are a few larger ones that have been found), but there is a special algorithm for testing mersenne primes. This method couldn't be applied to the 134 values after it

(this is why not all of the values between mersenne primes have been tested)

[boot2490]

I see. So there are likely those higher but not mersenne.
So what is your answer?

[sammyMaX]

Pretty easy. I got it!

[Xeda112358]

Well, I present you two answers...
Actually, only 1 since it was posted in IRC I will tell the other to Qwerty.55 if he wants, but I leave it up to you guys to find the other method

Theorem: If n and k are natural numbers where n is less than or equal to k, n|(k!+n)
Proof:
Let k and n be natural numbers
Since n is less than or equal to k, n is a factor of k!, so n|k!
Therefore, n|(k!+n)
☺
A corollary follows that {k!+2,k!+3,k!+4,...,k!+k} is a sequence of k-1 composite numbers

So, if k=133, we have a sequence of at least 132 composite numbers

[flyingfisch]

my solution was 10^120000 to 10^120000+132. (I wrote a prog in python to test that, since I was pretty sure there weren't many primes up there)