Really Big Numbers

Some really large numbers that have been featured in mathematical proofs:

Skewes' Number

The density of the prime numbers less than or equal to a number n is approximately equal to li(n), which is defined as the integral between 2 and n of dx/ln x. For relatively small values of n, this is an overestimate. In 1914, J. E. Littlewood proved that, as you go higher and higher, this formula switches from being an overestimate to an underestimate and vice versa an infinite number of times.

In 1933, S. Skewes, a student of Littlewood, proved (although he did assume the Riemann hypothesis) that the first switch occurs below e^e^e^79, which is about equal to 10^(10^(10^34)). This number is called Skewes' number. It is tremendously large, much larger than the number of particles in the universe or even a googolplex. Hardy called it "the largest number which has ever served any definite purpose in mathematics". Later on he took on the same problem without assuming the Riemann hypothesis, obtaining the even more enormous answer of about e^e^e^e^67, which is about equal to 10^10^10^10^29, and in 1955 obtained the "smaller" value of 10^(10^(10^967)). These numbers are upper bounds, though. Others have since lowered that bound to "only" 6.69×10 ³⁷⁰.

Graham's number

Graham's number is listed in the Guinness Book of Records as the highest number ever used in a mathematical proof. This number was derived by R.L. Graham in 1977 from a problem in Ramsay theory (a branch of combinatorics) which concerns bichromatic hypercubes. Normal exponential notation is not enough to express this number, so a special "arrow" notation, which was devised by Knuth in 1976, is used to represent this number.