On Prime Numbers

"The prime numbers are useful in analyzing problems concerning divisibility, and also are interesting in themselves because of some of the special properties which they possess as a class. These properties have fascinated mathematicians and others since ancient times, and the richness and beauty of the results of research in this field have been astonishing." (Carl H Denbow & Victor Goedicke, “Foundations of Mathematics”, 1959)

 “No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integers that refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are 'undecideable'. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)

“There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision.” (Don Zagier, “The First 50 Million Prime Numbers”, The Mathematical Intelligencer Vol. 0, 1977)

"Prime numbers have always fascinated mathematicians, professional and amateur alike. They appear among the integers, seemingly at random, and yet not quite: there seems to be some order or pattern, just a little below the surface, just a little out of reach." (Underwood Dudley, “Elementary Number Theory”, 1978)

“Prime numbers. It was all so neat and elegant. Numbers that refuse to cooperate, that don’t change or divide, numbers that remain themselves for all eternity.” (Paul Auster, “The Music of Chance”, 1990)

“If we imagine mathematics as a grand orchestra, the system of whole numbers could be likened to a bass drum: simple, direct, repetitive, providing the underlying rhythm for all the other instruments. There surely are more sophisticated concepts - the oboes and French horns and cellos of mathematics - and we examine some of these in later chapters. But whole numbers are always at the foundation.” (William Dunham, “The Mathematical Universe”, 1994)

"Prime numbers are the most basic objects in mathematics. They also are among the most mysterious, for after centuries of study, the structure of the set of prime numbers is still not well understood […]" (Andrew Granville, 1997)

"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." (Andreas Knauf, "Number Theory, Dynamical Systems and Statistical Mechanics", 1998)

“Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations.” (Marcus du Sautoy, “The Music of the Primes”, 1998)