An Affinity for Numbers

Ever since I was a kid, I have been fascinated by numbers and the seemingly endless possibilities they possess. We’ve all had our fair share of exposure to numbers, albeit differing substantially in range and volume. What the ordinary Joe deals with regularly falls under the umbrella of Natural numbers, numbers ranging from 1 to infinity having a denominator of 1. It’s fascinating to note that these natural numbers can further be classified under the Integers bracket and subsequently under the Rational and Real Numbers respectively.

Needless to say that there are myriad such categorizations possible when it comes to numbers, each rich with its scope of treatment and approach towards problem-solving. Another such category of numbers is known as Prime Numbers. Prime Numbers are the numbers consisting of only 2 factors that are zero and one. It is this particular category of numbers that has been intriguing mathematicians and aspiring students alike for centuries now.

A New Theorem on the Block

There has historically been a widespread desire to figure out an equation depicting the distribution of prime numbers, that may otherwise seem rather haphazard and unbecoming. One such theorem was first proposed in 1896 by Jacques Hadamard and Charles-Jean de la Vallée Poussin. The theorem, widely known as the Prime Number theorem provided a rudimentary treatment of the distribution these numbers exhibit across the number line.

The theorem provided a simple yet effective method of estimating the number of prime numbers less than or equal to a natural number on the number line. To understand the theorem, let’s consider a few examples. Consider the number 20. The list of prime numbers less than or equal to 20 includes 2,3,5,7,11,13,17 and 19. Hence there are a total of 8 prime numbers less than on equal to 20, represented mathematically as  Similarly, = 10 and = 25.

The Prime Number Theorem states that the number of primes less than or equal to a natural number is asymptotically given by  where n represents the said natural number and ln is the natural logarithm of the number. What’s interesting is that the theorem provides not the exact but the asymptotic distribution of these prime numbers. This has to do with the kind of distribution the numbers exhibit which admittedly isn’t linear by any means.

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Ever since I was a kid, I have been fascinated by numbers and the seemingly endless possibilities they possess. We’ve all had our fair share of exposure to numbers, albeit differing substantially in range and volume. What the ordinary Joe deals with regularly falls under the umbrella of Natural numbers, numbers ranging from 1 to infinity having a denominator of 1. It’s fascinating to note that these natural numbers can further be classified under the Integers bracket and subsequently under the Rational and Real Numbers respectively.

Needless to say that there are myriad such categorizations possible when it comes to numbers, each rich with its scope of treatment and approach towards problem-solving. Another such category of numbers is known as Prime Numbers. Prime Numbers are the numbers consisting of only 2 factors that are zero and one. It is this particular category of numbers that has been intriguing mathematicians and aspiring students alike for centuries now.

Density of Primes Less Than X

In simple words, graph tells us that the density of primes less than a particular natural number decreases as we move along the number line, eventually becoming almost 0 as we approach positive infinity. What’s notable here is that the accuracy of the theorem increases as we approach positive infinity, thereby making it that much more effective for calculating primes less than or equal to large numbers.

The Search for the Perfect Proof

The Prime Number Theorem has historically carried its share of baggage, originating right from its early days. The proof provided by the originators of the theorem, Hadamard and de la Vallée Poussin was deemed unsatisfactory owing to its heavy reliance on complex analysis since the theorem dealt with the distribution of natural numbers.

The following decades witnessed heated debates in the mathematical fraternity vis a vis a satisfactory elementary proof that could explain the intricacies of a theorem proving to be rather uncanny. Several independent proofs have since been provided that have aimed at improving the treatment of prime numbers with each passing iteration. Some of the notable ones have been posed by Atle Salberg (1948) and Helmut Maier (1985), that have relied on the properties of logarithms, opting to shun the rather tedious method involving imaginary numbers that was propositioned earlier. The trend has since then picked up with the mathematical fraternity arguing about the perfect proof and its existence to this day. (Newman’s Short Proof of the Prime Number)

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Carving the Way Towards a Prime Future

I’ve been following this journey of proving and reproving existing ideas for quite a while now. Admittedly, it excites and enthralls me. The idea of viewing existing ideas from a different perspective has always appealed to me. I’ve always felt that doing so opens up myriad new ways of inspection, that often lead to new and eye-opening results.

The emerging idea of always striving for a new and arguably better perspective in the mathematical fraternity is what will lead us heaps and bounds into the future, and carve out a future devoid of superfluous and self-serving notions of mathematical treatment. With luck, we may even end up with yet another theorem that sparks a debate transcending the fragility of time.

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