A Tale of the Primes — Part 2

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Prime numbers have fascinated humankind for millennia. The earliest investigation into the primes is evidenced by a bone discovered in the mountains of central equatorial Africa, which dates from 6500 B. C. and records all prime numbers between 10 and 20. The ancient Chinese, characterized by their practical approach to mathematics, developed a physical understanding of the prime numbers by as early as 1000 B. C. The ancient Greeks, distinguished by their abstract view on mathematics, set out to discover universal truths underlying the primes.

In the 4th century B. C., Greek mathematician Euclid, arguably the father of mathematics, showed that there are infinitely many primes – one of the first marvelous truths about the prime numbers discovered by men. Around the same time, the Greeks also proved that any number can be built by multiplying one or more prime numbers, thereby revealing the potency of prime numbers as fundamental building blocks in the world of mathematics.

Drawn to this potential power of primes, mathematicians since Euclid relentlessly studied these numbers. If the sciences resort to mathematics to describe patterns and orders in the universe, and if the prime numbers are the cornerstone of mathematics, wouldn’t the study of the primes bring new insights into the understanding of the world? However, the deeper mathematicians delve into the prime numbers, the more troubling mysteries they uncover.

A mathematician may begin the study of the primes by asking a simple question about the behavior of these numbers: How are the primes distributed in all numbers? Surely one can generate a distribution of primes by removing all non-prime numbers one by one, as demonstrated by the 3rd century B. C. Greek librarian Eratosthenes in his construction of tables of primes. But most mathematicians set out believing that there exists an order in the distribution of primes, a pattern that allows these numbers to be described by elegant formulae rather than tabulated by brute force.

One such mathematician was Pierre de Fermat (1601-1665). He proposed that raising 2 to the power of 2^N and adding 1 would generate a prime number. This formula, though unproven at the time, inspired mathematicians to construct a formula that describes a specific class of prime numbers. The French monk Marin Mersenne, for instance, constructed a formula, 2^p-1 (with p being any prime number), that generates a class of prime numbers now known as the Mersenne primes. The hope was that by constructing formulae describing specific primes, one might eventually stumble upon the secret to constructing a formula describing prime numbers in general.

Fermat’s first prime-generating formula, eventually disproved by Euler, was nonetheless a steppingstone for German mathematician Karl Friedrich Gauss, who conjured up an approximate description of prime numbers in general. Gauss (1777-1855), often referred to as the “prince of mathematics,” brought forth revolutionary ideas to most fields of mathematics: geometry, arithmetic, algebra, analysis, and, of course, the prime numbers. In 1801, Gauss proposed that the number of primes distributed between 1 and n is roughly equal to n over the natural logarithm of n. This rule for the distribution of prime numbers was proved forty-one years after Gauss died, and is now known as the Prime Number Theorem.

The Prime Number Theorem never correctly describes the prime numbers. It is only an approximation. Take the number 10,000,000,000 for example; if we count the prime numbers below it, there are exactly 455,052,512. The Prime Number Theorem predicts that there will be 434,294,493. The error is 20,758,019, or 4.5 per cent. To make matters worse, the error, though small, appeared to be unpredictable.

Working to remedy this deficiency was Gauss’s most eminent successor, Bernhard Riemann. Riemann (1826-1866) studied under Gauss and did important work in geometry, complex analysis, and mathematical physics, which ultimately led to Einstein's Theory of Relativity. Riemann was attempting a proof to guarantee that the per cent error of the Prime Number Theorem would get smaller as the numbers grew larger. What he discovered was what is now known as the Riemann zeta function. The Riemann zeta function indeed supplied the missing ingredient in Riemann’s proof, but there is something more peculiar about it. As Riemann began observing some prime numbers through the function, he saw that the primes would always order themselves in a perfect straight line. If this was true for all prime numbers, Riemann would have found a pattern that described the prime numbers exactly, as opposed to Gauss’s never-exact Prime Number Theorem.

Excited about this breakthrough, Riemann shared his idea in the 1859 paper On the Number of Primes Less Than a Given Magnitude to the Berlin Academy. As the mathematical community slowly came to grips with Riemann’s perspective on prime numbers, the brilliance of this new approach became evident. Riemann’s proposal of ordering the primes with his zeta function became known as the Riemann Hypothesis, which had set the course for the study of prime numbers to this day.

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