**ch3cooh**

For context, I'm making an activity for high school students in an afterschool math program. The activity is aimed at estimating, understating, and working with really big numbers, and this part of the program (4 sessions) is focused on bijections and infinties, with a theme of how mathematicians think about topics that are too large or abstract to handle with intuition, and for how mathematicians make choices that shape what the mathematics we study looks like.

For this very first activity in the first session, in addition to many mathematical examples, I want to include some examples from 'experiencial life' and things in the middle (like the limits of modern computation.)

Many sites say that the largest known prime is "2^57,885,161 − 1, a number with 17,425,170 digits." Given this and well known research about the density of primes, I think it's at least possible to estimate the number of primes between 1 and 2^57,885,161 − 1. But I don't know how to do this myself. I really want the answer to this one (the order of magnitude at least), but I've got lots of these, and I'd ideally like more cool ones. :)

I would greatly appreciate

A) Suggestions for additions to this list

B) Estimations for new additions or for the items listed below with question marks

C) Corrections to this list if you think one of my listed estimations is at the wrong order of magnitude

D) An estimate for the primes question in particular.

I asked this of "Ask Metafilter" a few minutes ago, and have already gotten back some amusing answers:

http://ask.metafilter.com/275591/Giant-numbers-Ex-how-many-primes-are-smaller-than-the-largest-known

THANKS!

----------------------------------------------------------------

I HAVE NO IDEA, BUT I'D LIKE TO USE THEM IF I CAN FIGURE OUT AN APPROXIMATION:

* The approximate number of primes between 0 and 2^57,885,161 − 1 (???)

* If two 'opponents' at Go are actually collaborating to make the game as long as possible, approximately how many turns might the game last (on a 19x19 board)? (???)

* The approximate number of different chess games for which the board never returns to a previously experienced state. (???)

* The approximate number of atoms in everything living on the earth (???)

I THINK THESE ARE RIGHT:

* The approximate circumference of the earth in miles (~10^4)

* The approximate number of sheets of paper in a stack as tall as the height of the Transamerica Pyramid in SF (~10^6)

* The approximate number of seconds in a human life (~10^9)

* The approximate number of people in the world right now (~10^10)

* The approximate number of neurons in a human brain (~10^11)

* The approximate number of seconds between now and when a dinosaur last lived (~10^15)

* The approximate number of ants in the world right now (~10^16)

* The approximate number of calculations that can be done by the world's current fastest computer (The Tianhe-2 supercomputer) in 1 minute (~10^18)

* The approximate number of stars in the universe (~10^30)

* The approximate number of atoms in the earth and everything on it (~10^50)

* The approximate number of different ways a deck of 52 cards might be shuffled (~10^68)

* The approximate predicted number of atoms in the universe (~10^80)

* The approximate number of rabbits that would exist in the universe after a year if, starting with 2 rabbits, ever day, every pair of rabbits alive mate and produce 6 offspring, and no rabbits ever die. (~10^219)

* The approximate value of the largest known prime number (~10^17425169)

See more at my ask MetaFilter page.

For this very first activity in the first session, in addition to many mathematical examples, I want to include some examples from 'experiencial life' and things in the middle (like the limits of modern computation.)

**The activity is not to calculate these quantities, but rather to estimate them to the extent that they can be ordered least to greatest.**Many sites say that the largest known prime is "2^57,885,161 − 1, a number with 17,425,170 digits." Given this and well known research about the density of primes, I think it's at least possible to estimate the number of primes between 1 and 2^57,885,161 − 1. But I don't know how to do this myself. I really want the answer to this one (the order of magnitude at least), but I've got lots of these, and I'd ideally like more cool ones. :)

I would greatly appreciate

A) Suggestions for additions to this list

B) Estimations for new additions or for the items listed below with question marks

C) Corrections to this list if you think one of my listed estimations is at the wrong order of magnitude

D) An estimate for the primes question in particular.

I asked this of "Ask Metafilter" a few minutes ago, and have already gotten back some amusing answers:

http://ask.metafilter.com/275591/Giant-

THANKS!

----------------------------------------

**The list so far:**

I HAVE NO IDEA, BUT I'D LIKE TO USE THEM IF I CAN FIGURE OUT AN APPROXIMATION:

* The approximate number of primes between 0 and 2^57,885,161 − 1 (???)

* If two 'opponents' at Go are actually collaborating to make the game as long as possible, approximately how many turns might the game last (on a 19x19 board)? (???)

* The approximate number of different chess games for which the board never returns to a previously experienced state. (???)

* The approximate number of atoms in everything living on the earth (???)

I THINK THESE ARE RIGHT:

* The approximate circumference of the earth in miles (~10^4)

* The approximate number of sheets of paper in a stack as tall as the height of the Transamerica Pyramid in SF (~10^6)

* The approximate number of seconds in a human life (~10^9)

* The approximate number of people in the world right now (~10^10)

* The approximate number of neurons in a human brain (~10^11)

* The approximate number of seconds between now and when a dinosaur last lived (~10^15)

* The approximate number of ants in the world right now (~10^16)

* The approximate number of calculations that can be done by the world's current fastest computer (The Tianhe-2 supercomputer) in 1 minute (~10^18)

* The approximate number of stars in the universe (~10^30)

* The approximate number of atoms in the earth and everything on it (~10^50)

* The approximate number of different ways a deck of 52 cards might be shuffled (~10^68)

* The approximate predicted number of atoms in the universe (~10^80)

* The approximate number of rabbits that would exist in the universe after a year if, starting with 2 rabbits, ever day, every pair of rabbits alive mate and produce 6 offspring, and no rabbits ever die. (~10^219)

* The approximate value of the largest known prime number (~10^17425169)

See more at my ask MetaFilter page.