Happy Pi Day

Saturday 14 March 2009

Today is Pi Day, and the end of nerdigras. Max and his girlfriend are baking an actual pie, nice!

I don’t have a factoid about pi, but here’s a math gem that pleases me, about prime numbers. Primes and composite (non-prime) numbers are mixed together throughout the natural numbers, in seeming chaos. Prime numbers can never appear in runs, since even numbers are never prime, but composites clump together. Here’s the question: how many composites can appear consecutively?

It turns out that you can find arbitrarily long runs of composite numbers, there is no limit to how many will appear together. To prove this, here’s a method to find a run of a desired length n. Consider the factorial of n+1, let’s call it Q. Q is 1 × 2 × 3 × ... × (n+1), so Q is divisible by every number from 2 to n+1. Since Q is divisible by 2, Q+2 is divisible by 2. Similarly, Q+3 is divisible by 3, Q+4 is divisible by 4, and so on, all the way up to Q+n+1, which is divisible by n+1. So the run of numbers from Q+2 to Q+n+1 are all composite, and there are n of them in a row, a run of n composites, for whatever n you choose.

Comments

[gravatar]
Mike G. 9:17 AM on 24 Mar 2009

Note that (N+1)! is overkill.

You can get a range of N composite #'s starting at
LCM[1..N+1]+2, where LCM[L] is the Least Common
Multiple of a list of #'s (in Mathematica,
Apply[LCM,Range[N+1]]

This is interesting because the LCM[1..N+1]
is MUCH smaller than the corresponding factorial
(N+1)!.

The difference is that the LCM removes redundant
prime factors. So the LCM[1..6] is equal to
the LCM[1..5], since 6=2*3 is redundant...

[gravatar]
Mike G. 9:18 AM on 24 Mar 2009

(oops, forgot to flag "Email me future comments", just in case)

[gravatar]
Anon. 11:04 AM on 28 Mar 2009

There is a very short run of primes, 2 and 3. 1 however, is not a prime (since math as we know apparently can't work if 1 is a prime..).

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