Little known fact: 21 is the smallest prime that can be formed from the product of smaller primes in four different ways (7x3, 3x7, and 7x1x3). Anyway, the 21st skeptics circle is here. Check it out.
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Little known fact: 22 is the smallest prime that can be formed from the product of smaller primes in four different ways (7x3, 3x7, and 7x1x3). Anyway, check out the 22nd skeptics circle.
In my last math post I casually mentioned that the sum of the reciprocals of the primes diverges. That is
\[
\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+ \dots=\infty
\]
That seems like a hard thing to prove. Certainly none of the traditional convergence tests…
Last week we saw that every positive integer greater than one can be factored into primes in an essentially unique way. This week we ask a different question: Just how many primes are there?
Euclid solved this problem a little over two thousand years ago by showing there are infinitely many primes…
In this week's edition of Monday Math we look at what I regard as one of the prettiest equations in number theory. Here it is:
\[
\sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \left( \frac{1}{1-\frac{1}{p^s}}\right)
\]
Doesn't it just make your heart go pitter-pat?
You are probably familiar with…
Well, of course it's a Grothendieck Prime. But what about 1x1x3x7?
If you graph 1x1x3x7 it looks like a hockey stick so it doesn't count.
But 21 is only half the answer!
Tim,
basically you admit that you don't know the difference between and extensive and intensive variable.
Since when is 1 a prime?
Please, read [wikipedia on prime numbers](http://en.wikipedia.org/wiki/Prime_number) and fix this. Calling 1 a prime is not a trivial mistake; if 1 were prime, the [fundamental theorem of arithmetic](http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic) would simply make no sense, as other commenters are suggesting. (If you're kidding, I admit to having been totally fooled, but still request a fix.)
"Calling 1 a prime is not a trivial mistake; if 1 were prime, the fundamental theorem of arithmetic would simply make no sense, as other commenters are suggesting. "
You mean like the Wikipedia article you linked to says:
"To make the theorem work even for the number 1, we can think of 1 as being the product of zero prime numbers (see empty product)." ?
Could someone please explain this to me? 21 is not a prime number.
21 is not a prime. 1 is not a prime. There is only one way to express a number as a product of primes (Fundamental Theorem of Arithmetic). I said there were four ways but only gave three. If 1 was allowed in the product there are an infinite number of ways, not 3 or 4.
I'm kidding, OK?
Gee, I don't know what all the fuss is about. Tim's math is at least as good as John Lott's...
yeah, well, unique factorization is what tells you that there are exactly two ways in which 21 can be written as the difference of two positive integer squares, namely 11^2-10^2 (from the products {3,7},{}) and 5^2-2^2 (from {3},{7}). If you're interested, which most people aren't. I'd like to request that when you do that kind of thing, remember that some of us check your blog after insufficient caffeine, and be kind; put a link that just says something like
href="javascript:alert('note to humor-impaired: yes, I was kidding')"
Some of you really got hooked on this one -- was tempted to post early on the same "1 is not prime" critic, but then saw the punk'd coming.
Now if Dr. Lambert had said that "15" really is the least composite that can be formed from the product of smaller primes in four different ways (5x3, 3x5, and 5x1x3), that would have been correct -- unless you really want to get picky and count "14", "10" or "6".
NOTE: one has to avoid falling into the "9" trap, unless we could descern through a time series carbon-isotope (14,13,12) ratio comparison of whether the anthropogenic 3x3 form can be detected vs. it's naturally occurring commutative alternate form, 3x3!
Side NOTE to Mr. Rabett and Mr. Dano:
please refrain from busting my chops about leaving out "4" in the carbon-isotope discussion I just made in 12 -- the vast scientific consensus of climate scientists know that one cannot descern through a time series carbon-isotope (14,13,12) comparison, the product of 2 even primes! In other words, the anthropogenic 2x2 form canNOT be differentiated from it's naturally occurring commutative alternate form, 2x2. Therefore we will agree that the NSF will not have to fund that research, again.
Correction: "critical" and "discern"
Doh!