Stumbling through mathematics: no one has Skewes' number

I once saw a bumper sticker that read EVERYONE HAS AVAGADRO'S NUMBER. Avagadro's number is the number of molecules in one mole of a substance, and written in scientific noation is approximately equal to 6.0225 × 10²³. In standard notation, Avagadro's numbers is approximately:

602,250,000,000,000,000,000,000

By contrast, no one can write a Skewes' number in standard form. There are too many digits.

So, why is a Skewes' number so interesting (to mathematicians, anyway)?

A little background information is necessary here. A googol is the number 10¹⁰⁰, which is 10 raised to the one hundred power. In standard form, a googol is 1 followed by 100 zeros, or:

100000000000000000000
00000000000000000000
00000000000000000000
00000000000000000000
00000000000000000000

One function we need to know about is called the prime counting function π(x) . This function is the number of prime numbers less than or equal to x. Some examples would be π(1) = 0, since there are no prime numbers of 1 or less; π(6) = 3, since 2, 3 and 5 are all prime numbers less than or equal to 6; π(13) = 6 since 2, 3, 5, 7, 11, and 13 are all prime numbers less than or equal to 13; and so on. (Note that π(x) is not the famous irrational number π ≈ 3.1415926...)

Another function that we must have to understand what a Skewes' number is about is the logarithmic integral li(x). This function is defined as:

li(x) = ∫₀^xdt/ln(t)

where ln(t) is the natural logarithm of t.

Below is a plot of π(x) and li(x) versus x:



Notice that li(x) (the red line) is always above π(x) (the blue line) Or so it would appear.

Actually, appearances can be deceiving, and in this case they are. For eventually li(x) and π(x) cross each other.

So where does the first crossing occur? In 1933 Samuel Skewes (Derbyshire 2004) showed that the first such crossing* will happen before x is equal to the number:

e^ee79

This number is called the first Skewes' number, or Sk₁. The symbol e is the base of the natural logarithm, which is an irrational number and approximately equal to 2.718281828459045.... Sk₁ should be read as, "e raised to the power of e raised to the power of e raised to the power of 79" (such a stacking of exponents is sometimes called a "power tower"). Compare this gigantic number with the now puny-looking googol!

How many digits does Sk₁ have? Remember that the googol is a 1 followed by 100 digits. According to Derbyshire, the first Skewes' number has 10^{ten billion trillion trillion} digits. Just try writing that number in standard form!

Of course, it is easy to propose even bigger numbers than Sk₁. The trouble is that such a number is a "made-up" or invented number. What makes Skewes' number so interesting is the way it was "discovered". It is the largest number (so far) to come about as a result of a mathematical proof**.

Recent advances in mathematics have reduced the first crossing of li(x) and π(x) to a much, much smaller number (on the order of 10³¹⁶). Nevertheless, the Skewes' number remains one of the more fascinating numbers in mathematics.

(* - assuming the Riemann hypothesis is true.)

(** - if the Riemann hypothesis is false, then a far larger Sk₂, or the second Skewes' number is the result. However, most mathematicians believe the Riemann hypothesis is true.)