So, this is the starting point of a series of serious large numbers. That is because, these numbers are somewhat famous in mathematics.
These numbers are the upper-bound solutions to a prime-counting function. The numbers are named after Stanley Skewes (1899-1988), a South African mathematician. The lower bound is assured when Riemann hypothesis is assumed to be true. This bound was found in 1933.
The problem states that the smallest number n in which the prime counting function, π(x) is greater than logarithmic integral, li(x).
The prime counting function, denoted π(x), is the function that returns the number of primes less than or equal to x.
For example, π(7) is equal to 4, as there are 4 primes less than or equal to 7, π(100) is equal to 25, as there are 25 primes less than or equal to 100, etc.
Nowadays, calculating number of primes looks simple, but actually complex. There are a lot of processes behind it. However, the logarithmic integral function also serves as a good approximation to prime counting function.
Normally, π(x) < li(x) for x greater than 2, but, there are bounds that π(x) > li(x). This proof was shown in 1914, by John Edensor Littlewood, according to Saibian.
In 1933, Skewes proved that π(x) > li(x) occurs at e^e^e^79, where e is a transcendental number defined as the the sum of the reciprocal of factorials from 0 to infinity (note that 0! is equal to 1).
Let us show that the sum of the reciprocal factorials converges to e below, step-by-step, up to 8 decimal places:
1÷(0!) = 1
1 + 1÷(1!) = 2
2 + 1÷(2!) = 2.5
2.5 + 1÷(3!) = 2.66666667
2.66666667 + 1÷(4!) = 2.70833333
2.70833333 + 1÷(5!) = 2.71666667
2.71666667 + 1÷(6!) = 2.71805556
2.71805556 + 1÷(7!) = 2.71825400
2.71825400 + 1÷(8!) = 2.71827877
2.71827877 + 1÷(9!) = 2.71828153
2.71828153 + 1÷(10!) = 2.71828180
2.71828180 + 1÷(11!) = 2.71828182
After 11 factorials, the result converges to 2.71828182, which agrees with e, correct to 8 decimal places.
The first value is obtained when assuming Riemann Hypothesis is true. Riemann Hypothesis was proposed in 1859, via a paper