Skewes’ Number
Ever since humanity started grasping the least of arithmetical concepts, one class of numbers has always been at the forefront – the primes. Straightforward to define but elusive to catch, the primes have been notoriously responsible for some of the hardest problems in mathematics, puzzling the best of mathematicians across generations.
“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.” ~ Leonhard Euler
The Prime Counting Function
Perhaps the most famous of those problems has been this – given a number n, find a function that outputs the number of primes less than or equal to n. The required function is represented as π(n).
For example –
π(10) = 4 (2, 3, 5, 7)
π(17) = 7 (2, 3, 5, 7, 11, 13, 17)
Finding a closed form (exact) expression for π(n) has long been one of the most coveted problems in mathematics. While no one has been able to come up with an exact expression, we’ve discovered some mighty good approximations!
Approximations to the Prime Counting Function
First, a word on what we mean by approximation here. When we say a function f approximates a function g, it implies that for large enough values of n, f(n) and g(n) are almost identical, i.e., the percentage difference between their values can be safely ignored for practical purposes (the interested reader may further look up Big O notation). That is, f(n) ~ g(n) for large n.
Now, we present the two most well-known approximations to the Prime Counting Function.
- N/lnN: Here, ln is the natural logarithm, i.e, logarithm to the base e. This is the simplest approximation to the prime counting function. Surprisingly, this straightforward expression yields values in excellent agreement with π(n). Also, the expression usually outputs values lesser than π(n), thereby serving as an informal lower bound for π(n).
- Li(N): The second approximation is the logarithmic integral. This function is defined as
Graphically, it is the amount of area enclosed between the curve 1/ln(x) and the x-axis, with x ranging from 2 to N.
Unlike the previous expression, Li(N) normally outputs a value greater than π(n), thereby serving as an informal upper bound for π(n).
Now comes the interesting part. Notice the use of the words “normally” and “informal” in the above statements? They’ve been employed deliberately. Let me elaborate on the importance of the two words —
- “Normally”: It means that the statements hold true, without contradictions, for the typical range of values we concern ourselves with for practical purposes.
- “Informal”: But there is no mathematical proof that these statements hold true everywhere, i.e., all the way up to infinity.
So, restricting ourselves to discussion of Li(N), is it possible that it falls below π(N) for some large values of N? If yes, what is that value of N? That, precisely, is the main agenda of this article!
Littlewood’s Result
For many years post the discovery of Li(n) as an approximation to π(n), the following question remained unanswered, troubling the best minds of the time –
Is Li(n) > π(n) for every possible n? Or does the inequality flip somewhere down the line?
People had calculated Li(n) values as high as they possibly could, but still not even a single counter example was stumbled upon. Probably, many people would have assumed the inequality to hold undeterred for all values of n.
Finally, the question was resolved by one of the leading British mathematicians of the era, J.E. Littlewood — and resolved in the affirmative! Littlewood proved the inequality indeed flips signs for some large enough N, i.e., there exists an N such that Li(N) < π(N). In fact, Littlewood proved much more. He showed that the inequality flips signs infinitely many times! That is, if we looked high enough, we’ll find infinitely many patches where Li(N) dominates π(N), and vice-versa. Quite some result indeed!
But, however glorious Littlewood’s result was, it lacked one thing – a bound for when the first flip would occur. The result was a classic example of an existential proof in mathematics – asserting that something exists without specifying any particular example of it, or even an algorithm to construct one.
So, even after Littlewood’s substantial advancement, the question remained – when does the inequality flip signs? Can we have some upper bound at least?
That, precisely, is what was left for Littlewood’s student, Skewes, to demonstrate!
Skewes’ Number
Stanley Skewes (1899–1988) was a South African mathematician who studied under Littlewood at Cambridge University. After initially obtaining a degree in civil engineering, he emigrated to England to study mathematics, and obtained a PhD in the same in 1938.
Skewes’ number is an upper bound for the flip discovered by Littlewood. That is, we can be certain that π(n) becomes greater than Li(n) for some number smaller than Skewes’ number!
To be theoretically rigorous, there’s not just one unique Skewes’ number. Rather, there are two of them, subject to the validity of the Riemann hypothesis. For the uninitiated, the Riemann hypothesis is the most coveted open problem in the entirety of mathematics, let alone number theory. While going into the details of the problem is beyond this article’s scope, the following understanding should suffice –
- First Skewes’ Number: Assuming the Riemann hypothesis to be true, Skewes, in 1933, showed that there must be a number below 10^(10^(10³⁴)) for which π(n) becomes greater than Li(n). Usually, this number is simply referred to as Skewes’ number. No wonder people were sceptical if the flip over indeed took place at all! Calculations at such gargantuan scale are out of bounds even for today’s mighty computers, let alone the devices available a century back.
- Second Skewes’ number: It would be another 22 years before Skewes could come up with a corresponding limit in case the Riemann hypothesis is false. His 1955 result proved that even if the Riemann hypothesis is false, the flip between π(n) and Li(n) has to happen somewhere below 10^(10^(10⁹⁶⁴)). That is the second Skewes’ number, clearly even more humongous than the first!
Conclusion
The great British mathematician (and Littlewood’s long-time collaborator), G.H. Hardy, described Skewes’ number as –
“[Skewes’ number is] the largest number which has ever served any definite purpose in mathematics.” ~ G.H. Hardy
More than anything else, this statement highlights the extent to which pure mathematicians can go to satisfy their intellectual curiosities. From a practical point of view, Skewes’ result doesn’t contribute anything substantial. The world would remain the same even if we were oblivious of it. Anyways, who’d want to do anything with a number that can’t even be written down completely!?
But then, pure mathematics hardly cares about its applicability. Analogous to art pieces, mathematical results like Skewes’ have a fan following of their own, however niche. Therein lies the beauty and purpose of pure mathematics — it exists to satiate the intellectual curiosity of a select group of people — across space and time!
