Subsequent occurrences are fairly common.
Brun’s and Meissel-Mertens ConstantsPrime numbers make an appearance in two fascinating constants we’ll discuss next.
Brun’s constant takes the form below — the sum of the reciprocal of all prime pairs.
The number converges extremely slowly, which is fascinating in itself.
The denominator of these fractions decreases quickly — the problem is that prime pairs are relatively common even for very large numbers.
In fact, Brun’s constant is not known.
It was estimated to be ~1.
902160583104 using all twin primes up to 10¹⁶ (that is, 10 thousand billion), but is only proven to be less than 2.
347.
Returning to Google: the company famously bid Brun’s constant ($1,902,160,540) in the auction for the patents of bankrupt Nortel corporation.
A subsequent bid of $2,614,972,128 mirrored the Meissel-Mertens constant (approximately 0.
26149…).
The Meissel-Mertens constant is also referred to as the prime reciprocal constant, due to its definition below:This number again is not known exactly, though converges much quicker than Brun’s constant.
The Golden Ratio and Fibonacci NumbersStudied since Euclid’s time, the Golden ratio plays an important role in geometry, and has many geometric representations.
The last spiral image is called the Fibonacci spiral, and represents a fascinating property of the Golden Ratio: taking the ratio of successive Fibonacci numbers converges to the Golden Ratio, which we’ll discuss below.
A Fibonacci number takes the form F(n) = F(n-1) + F(n-2).
The beginning of the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
A few fascinating properties of Fibonacci numbers: every positive integer can be written as a sum of Fibonacci numbers where any one number is used only once.
This is maybe intuitive when you consider the density of front part of the sequence relative to later parts.
Fibonacci numbers are the sum of the diagonals of Pascal’s triangle (constructed with each successive layer being the sum of the two numbers above).
The Fibonacci numbers are the sums of the “shallow” diagonals (shown in red) of Pascal’s triangle.
As mentioned above, the Golden ratio converges when you take the ratio of successive Fibonacci numbers.
For example, 89/55 = 1.
61818, which is close to the true value of 1.
61803.
The Golden ratio can be defined exactly using the irrational number root 5.
The Golden ratio can also be expressed as a continued fraction, so we can use or e-generating algorithm above to generate digits of the golden ratio.
Specifically:phi = ''for digit in continued_fraction(lambda k: 1, lambda k: 1, 500000): phi += str(digit)You’ll note that running the code above takes a while compared to the e code.
The 500,000 asks the algorithm to return 500k digits, without respect to how many iterations of the continued fraction are required.
The golden ratio continued fraction converges slowly, which makes some sense: growing numbers in the denominator allows the recurring additive fractions smaller.
ConclusionThere are many more constants we could discuss, but this article is already a bit of a rambling exploration!.We didn’t get to the first sophomore's dream constant (the area bounded by curve x^x between 0 and 1, so named for the incorrect, but oh-so-convenient identity (x+y)^n = x^n + y^n that would make such an integral much easier to solve); values of the Gamma and Reimann zeta functions that appear in statistics; or any of the other constants that make appearances in stochastic random walks and other analytical applications.
Main take-away from this article: math is super cool!.And Google has a good sense of humor!.. More details