An artist’s stab at maths
- What are the numbers above?
- Some prime numbers I found at Wikipedia.
- That reminds me. I got a letter from a student of mine.
- A maths student?
- No she was more into art. Anyway, she has found a proof that amazes me.
- I am intrigued!
- She studied p1 * p2 * p3 * … * pn + 1 where pi is the ith prime number.
- I am lost!
- Let me give you an example. For n = 3 she multiplied the first three prime numbers and added one.
- What on earth for?
- 2 * 3 * 5 + 1 = 31.
- 31 is on the Wikipedia list.
- That is what she noticed! She proved that p1 * p2 * p3 * … * pn + 1 always gives a prime.
- I have some bad news. Euclid, who was not born yesterday, used that fact to show that there are an infinite number of primes.
- Really? I guess I should get out more often.
June 17th, 2010 at 2:46 am
Interesting. Primes are very neat: simple in concept but often the facts concerning primes are either mindblowingly difficult to prove and often thus far unproven.
Since I’m not match for many of the minds in this blog, let alone Euclid himself, I’d appreciate a critique of my layman’s description of why this is true.
Any non-prime number is by definition a composite of some quantity of primes.
A rudimentary test of primality is to use brute force…. [as an aside, Google is far easier than brute force]:
1) Test n for primality (example: 127)
2) Find an approximate value for ?n (example: 11 < ?127 ?n is a factor, it must be paired with some number <?n, would have by this point already have been tested.
OK so that’s all basic but serves as a foundation to why I think Euclid stated this.
So… Taking a look at the number 8, it factors out to 2³. 8+1 can’t possibly contain 2 as a factor because the number immediately prior to it “used up” the factor of 2. In this example, 9 is of course not a prime, allowable because while 8 used up the factor of 2, it does not touch 3.
Taking a look at the number 30, it factors out to 2*3*5. 30+1 must be prime because to test for primality we’d look at ?31 (some number n between 5 and 6). We need to test every prime from 2 to n but don’t need to bother testing any factors of 30 (2, 3, and 5). Well… 2, 3, and 5 are the only numbers we would have tested for 31.
Interesting. It’s far simpler to wrap my mind around it after Euclid did the hard work. I can’t imagine being bright enough to have come up with that from scratch as a completely new and fresh idea.
~~~
As an aside, it’s important to note that this in no way implies that every prime number -1 is a multiple of consecutive primes.
June 17th, 2010 at 2:47 am
I failed at grammar in my first paragraph.
June 17th, 2010 at 2:58 am
I failed at grammar in my second paragraph as well…. sigh.
Also, where you see oddly placed question marks, they’re often supposed to be the square root sign. I forgot that is not allowed here.
June 17th, 2010 at 3:30 am
“She proved that p1 * p2 * p3 * … * pn + 1 always gives a prime.”
I’d like to see how her proof proves that 2*3*5*7*11*13 + 1 = 30031 is prime.
(n<6 and n=11 do indeed give primes, but all other n up to at least 74 give composites.)
June 17th, 2010 at 4:17 am
For n=7, 2*3*5*7*11*13*17+1=510511, which is 19*97*277.
I fell into a trap of assuming that the initial problem, which stated that Euclid had proven the conjecture as fact, was absolutely true.
June 17th, 2010 at 8:27 am
Here’s Euclid’s proof described:
http://primes.utm.edu/notes/proofs/infinite/euclids.html
June 17th, 2010 at 9:54 am
Here’s Euclid in the classroom:
http://easyquestion.net/miscajitas/publicadas/atmdialogue.htm