In this article, we will cover the Skewes' numbers, two very large upper-bounds to a problem related to the distribution of primes, devised by Stanley Skewes. It may seem strange that these upper bounds originated from a problem involving prime numbers, as they are both vastly larger than the largest known prime number.
The prime-counting function, usually denoted with the Greek letter pi, returns the number of primes up to its input, while the logarithmic integral li(x) is the definite integral of 1/ln(x) from 2 to x. For most values of x, pi(x) < li(x). However, it was proven that the sign of pi(x) - li(x) changes infinitely many times, implying that there is some integer out there for which pi(x) is greater than li(x).
The first Skewes' number was the upper-bound for the first such integer assuming the Riemann hypothesis is true. It is exactly equal to e^(e^(e^79)). It is normally approximated at 10^(10^(10^34)), and sometimes more accurately to 10^(10^(8.852*10^33)).
e79 = 20382810665126687668323137537172632.374...
log10(e)*e79 = 8852142197543270606106100452735038.912894...
log10(e)*e79 + log10(log10(e)) = 8852142197543270606106100452735038.550679... (about 1033.9470483)
So the number of digits in the integer part of the first Skewes' number is approximately 3.55*108852142197543270606106100452735038 (about 10^0.550679 multiplied by 108852142197543270606106100452735038). We will never know any of the digits, as it becomes impossible to calculate the first digits of nontrivial exponential numbers around 10^10^10^10, which is still much smaller than 10^10^8.852142*10^33, and with a non-integer base there is no modular exponentiation trick for the last digits of the integer part as there is for integer powers.
Now, we will see how much larger the first Skewes' number is than a googolplex.
(10^10^100)(10^10^100) = 10^(2*10^100) = 10^10^100.3010299956... = 10^10^10^2.0013053928...
So multiplying a googolplex by itself does not get us to 10^10^10^2.002, let alone anywhere near Skewes' number. Now we will try multiplying a googolplex by itself twice, or cubing the googolplex:
(10^10^100)(10^10^100)(10^10^100) = 10^(3*10^100) = 10^10^100.4771212547... = 10^10^10^2.002067183...
Even multiplying by a googolplex again does not get us past 10^10^10^2.0025, let alone 10^10^10^33.947048. So now we will try the tenth, hundredth, thousandth, and googolth powers of the googolplex:
(10^10^100)^10 = 10^10^101 = 10^10^10^2.004321373...
(10^10^100)^100 = 10^10^102 = 10^10^10^2.008600171...
(10^10^100)^1000 = 10^10^103 = 10^10^10^2.012837224...
(10^10^100)^10^100 = 10^10^200 = 10^10^10^2.301029995...
Raising the googolplex to the googolth power still does not get us anywhere near 10^10^10^33.947048. The power that we will have to raise a googolplex to to get to approximately the first Skewes' number is: 3.55*108852142197543270606106100452734938. What?! That's barely smaller than the number of digits in Skewes' number! Well actually it is 1/10^100 of that number.
And, the power you would have to raise Skewes' number to in order to get the common approximation of 10^10^10^34 is about 2.8*101147857802456729393893899547264961. While 10^10^10^34 doesn't seem much larger than 10^10^10^33.947048, it is actually like taking an entire universe with 10^(10^(8.8521422*10^33)) particles, and imagining a larger multiverse that is exactly identical but each particle is a universe, then an omniverse where each particle is a multiverse, and continue for 10^10^1.1478578*10^33 steps!
The second Skewes' number is the upper bound to the same problem if the Riemann hypothesis is false. It is commonly approximated to 10^10^10^1000, or more accurately 10^10^10^963. It is actually closer to 10^(10^(10^963.5185)).
Now, using computers, we have been able to determine that the crossover is actually around 1.397162*10316. There is no specific integer known to satisfy pi(x) > li(x), and in fact the value in this paragraph is not even known to be the first crossing. The original upper-bounds are now merely footnotes in mathematical history, which now merely serve as examples of surprisingly large numbers arising from a serious mathematical proof.