• 3:43 :D﻿ .. 3:45 :l

• The definition of product can be expanded to include the product of one number.

Have you ever seen summation notation?

∑[n=1,3] n = 1+2+3.

We can have the summation of one number still make sense, though.

∑[n=1,1] n = 1. The sum of one number is the number itself.

Similarly, we have a notation for products.

∏[n=1,3] n = 1*2*3

So ∏[n=1,1] n = 1. We have the product of 1 number - the number itself.

So primes are the product of one prime, much﻿ like how 1 is the product of zero primes.

#### Video Responses

• It would make sense to me to say, in the theorem, that for each of these products 1 must be included exactly once.

If we﻿ do that we can make 1 = 1. 2 = 1*2. 15 = 1*3*5 etc The products are still unique and 1 is a prime number.

• 1﻿ is all alone. so lonely.... *plays sad music*

• Thanks for the info, I was having a bit of fun on how our views of science and the like change as our understanding grows through out time.﻿ :)

• Well, historically, 1 was only ever considered a prime by a majority of mathematicians for at most 250 years (which is relatively short in the history of mathematics).

It is likely that the general consensus of﻿ the mathematical community that 1 should NOT be considered a prime was reached around the same time that Pluto was discovered (or possibly earlier!).

• First they get ride of Pluto as a planet, now they get rid of the number one as﻿ a prime number. My world is every changing. :) how am I to keep up?

• You just broke﻿ every encryption algorithm.

• Just 11. It is﻿ a product with one factor.

• Exactly. A power is just a product of the same number.

Now an empty product doesnt alter the value if you multiply by it. It is neutral because there are no factors in it.

The neutral element of multiplication is 1 as in﻿ x * 1 = x, so the empty product (or x⁰) is 1.

Note that 0⁰ is undefined though.

• So I guess 2⁰ and 3⁰ are considered the same way to﻿ express 1 as a unique product of primes?