Prime numbers and spirals

Prime numbers, yes. You might have been introduced to these numbers in elementary school but some of their complexities leave us bewildered and have gained the fascination of many mathematicians including mathematics greats like Gauss, Euclid and Riemann making them one of the most sought after bases of studies in mathematics.

But what is so mysterious about these seemingly simple numbers?

We are all well aware about the uniqueness of prime numbers about having only 2 factors: 1 and the number itself , but have we ever thought how widely they occur in a given set of numbers? 

How many primes are there less than 10? 5

Less than 100? 25

How about a million now?

You can’t just simply list all the prime numbers like you could before. So how about figuring out a trend in these? 

That could work, but how exactly?

Let’s look at how Gauss went about tackling this challenge.

He picked up a pen and kept on writing numbers while calculating the  numbers of primes for each set of numbers until he figured the shape of their distribution and arrived at the solution. It might seem bizarre to many but the answer can be found hidden in so many things around us: flowers, storms, galaxies, and fossils. The answer is hidden in their shapes, that is a logarithmic spiral.

Logarithmic Spirals in Nature

The shape of the logarithmic spiral provides a clear understanding of the distribution of primes and is well attributed to it. As we move outwards in the spiral starting from its center, our distance from the center keeps on increasing. This is a representative of the density (Number of primes / Search Size) of distribution of the primes.

Distribution of Primes

There are 25 primes less than 100 ( density- 25%), 168 primes less than 1000( 16.8%), 1229 primes less than 10,000( 12.29%) , and this goes on. But what can be observed is, how the density of primes in the search size decreases as the length of the  search size increases. Or how the further we go in terms of the integers we consider, the number of primes per the search size thins out.

Expression for the number of primes

Keeping in mind that the number of primes will follow a logarithmic shape, Gauss was able to arrive at a formula for the density of the primes in a search size (x) i.e. 1/ln(x).

Now density(d)= No. of primes in the search size / Search size

So,

No. of primes = (Density) X (Search size) = x/ln(x)

This expression , although not completely accurate, does a fairly excellent job to help us estimate the number of primes. Who would’ve thought that the answer, a mystery of its time, remained in something we get to see everyday. 

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