"A perfect number is that which is equal to the sum of its own parts." (Euclid, "Elements", cca. 300 BC)
"A prime number is one (which is) measured by a unit alone." (Euclid, "Elements" Book VII, cca. 300 BC)
"If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect." (Euclid, "Elements", cca 300 BC)
"Numbers prime to one another are those which are measured by a unit alone as a common measure." (Euclid, "Elements" Book VII, cca 300 BC)
"Uneven numbers are the god’s delight" (Virgil, "The Eclogues", cca. 40 BC)
"Why do we believe that in all matters the odd numbers are more powerful […]?" (Pliny the Elder, "Natural History", cca. 77-79 AD)
"Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little." (Nicomachus of Gerasa, "Introductio Arithmetica", cca. 100 AD)
"Numbers are called prime which can be divided by no number; they are seen to be not ‘divisible’ by the monad but ‘composed’ of it: take, for example, the numbers live, seven, eleven, thirteen, seventeen, and others like them. No number can divide these numbers into integers. So, they are called `prime,' since they arise from no number and are not divisible into equal proportions. Arising in themselves, they beget other numbers from themselves, since even numbers are begotten from odd numbers, but an odd number cannot be begotten from even numbers. Therefore, prime numbers must of necessity be regarded as beautiful." (Martianus Capella, cca. 400 AD)
"Number is divided into even and odd. Even number is divided into the following: evenly even, evenly uneven, and unevenly uneven. Odd number is divided into the following: prime and incomposite, composite, and a third intermediate class (mediocris) which in a certain way is prime and incomposite but in another way secondary and composite." (Isidore of Seville, Etymologies, Book III, cca. 600)
"[…] in the science of numbers ought to be preferred as an acquisition before all others, because of its necessity and because of the great secrets and other mysteries which there are in the properties of numbers. All sciences partake of it, and it has need of none." (Boethius, cca. 6th century)
"We should not leave unmentioned the principal numbers […] those which are called ‘perfect numbers’. These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason **, ***, and **** are called perfect numbers." (Hrotsvit of Gandersheim, "Sapientia", 10th century)
"There is divinity in odd numbers, either in nativity, chance, or death." (William Shakespeare, "The Merry Wives of Windsor", 1602)
"For any number there exists a corresponding even number which is its double. Hence the number of all numbers is not greater than the number of even numbers, that is, the whole is not greater than the part." (Gottfried W Leibniz, "De Arte Combinatoria", 1666)
"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)
"It will seem not a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of the numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful." (Leonhard Euler, "Specimen de usu observationum in mathesi pura" , Novi Commentarii academiae scientiarum Petropolitanae 6, 1756/57)
"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. […] The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl Friedrich Gauss, "Disquisitiones Arithmeticae", 1801)
"Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers" (Carl F Gauss, "Disquisitiones arithmeticae" ["Arithmetical Researches"], 1801)
"The difference of two square numbers is always a product, and divisible both by the sum and by the difference of the roots of those two squares; consequently the difference of two squares can never be a prime number." (Leonhard Euler, "Elements of Algebra", 1810)
"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)
"Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories." (George B Mathews, "Theory of Numbers", 1892)
"Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them." (Henri Poincaré, "Annual Report of the Board of Regents of the Smithsonian Institution", 1909)
"The Perfect numbers are also like the virtues, few in number; whilst the other two classes are like the vices - numerous, inordinate and indefinite." (W Wynn Westcott, "Numbers: Their Occult Power and Mystic Virtues", 1911)
"The theory of numbers is unrivalled for the number and variety of its results and for the beauty and wealth of its demonstrations. The Higher Arithmetic seems to include most of the romance of mathematics." (Louis Mordell, 1917)
"The theory of numbers is the last great uncivilized continent of mathematics. It is split up into innumerable countries, fertile enough in themselves, but all the more or less indifferent to one another’s welfare and without a vestige of a central, intelligent government. If any young Alexander is weeping for a new world to conquer, it lies before him." (Eric T Bell, "The Queen of the Sciences", 1931)
"The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found it in it a mysterious attraction impossible to resist." (Georg H Hardy, 1915)
"Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them." (Henri Poincaré, "Annual Report of the Board of Regents of the Smithsonian Institution", 1909)
"The Perfect numbers are also like the virtues, few in number; whilst the other two classes are like the vices - numerous, inordinate and indefinite." (W Wynn Westcott, "Numbers: Their Occult Power and Mystic Virtues", 1911)
"The theory of numbers is unrivalled for the number and variety of its results and for the beauty and wealth of its demonstrations. The Higher Arithmetic seems to include most of the romance of mathematics." (Louis Mordell, 1917)
"The theory of numbers is the last great uncivilized continent of mathematics. It is split up into innumerable countries, fertile enough in themselves, but all the more or less indifferent to one another’s welfare and without a vestige of a central, intelligent government. If any young Alexander is weeping for a new world to conquer, it lies before him." (Eric T Bell, "The Queen of the Sciences", 1931)
"The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found it in it a mysterious attraction impossible to resist." (Georg H Hardy, 1915)
"The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. […] Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers." (Godfrey H. Hardy, "The Theory of Numbers", Nature 1922)
"The mystery that clings to numbers, the magic of numbers, may spring from this very fact, that the intellect, in the form of the number series, creates an infinite manifold of well distinguishable individuals. Even we enlightened scientists can still feel it e.g. in the impenetrable law of the distribution of prime numbers." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1927)"Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number - there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method." (John von Neumann, "Various techniques used in connection with random digits", 1951)
"On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory." (Kurt Gödel, 1951)
"The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. ‘Perfect numbers’ certainly never did any good, but then they never did any particular harm." (John E Littlewood, "A Mathematician’s Miscellany", 1953)
"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 integerst hat 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 ‘undecidable’. 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)
"[A] sequence is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution." (J. N. Franklin (1962)"[…] random numbers should not be generated with a method chosen at random. Some theory should be used." (Donald E. Knuth, "The Art of Computer Programming" Vol. II, 1968)
"The generation of random numbers is too important to be left to chance." (Robert R. Coveyou, 1969)
"[…] there is no apparent reason why one number is prime and another not. To the contrary, upon looking at these numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation." (Don Zagier, "The First 50 Million Prime Numbers", The Mathematical Intelligencer, Volume 0, 1977)
"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)
"What will prove altogether remarkable is that some very simple schemes to produce erratic numbers behave identically to some of the erratic aspects of natural phenomena." (Mitchell Figenbaum, "Universal Behavior in Nonlinear Systems", 1980)
"Some order begins to emerge from this chaos when the primes are considered not in their individuality but in the aggregate; one considers the social statistics of the primes and not the eccentricities of the individuals." (Philip J Davis & Reuben Hersh, "The Mathematical Experience", 1981)
"Maybe some simple combination of a dozen or so primes in fact yield an odd perfect number!" (Stan Wagon, "The evidence: perfect numbers", Mathematical Intelligencer 7(2), 1985)
"It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means.'' (Rob C Vaughan, 1990)
"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)
"[Number theory] produces, without effort, innumerable problems which have a sweet, innocent air about them, tempting flowers; and yet…number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort!" (Barry Mazur, "Number Theory as Gadfly", The American Mathematical Monthly, Volume 98, 1991)
"To me, that the distribution of prime numbers can be so accurately represented in a harmonic analysis is absolutely amazing and incredibly beautiful. It tells of an arcane music and a secret harmony composed by the prime numbers." (Enrico Bombieri, "PrimeTerritory", The Sciences, 1992)
"[…] number theory […] is a field of almost pristine irrelevance to everything except the wondrous demonstration that pure numbers, no more substantial than Plato’s shadows, conceal magical laws and orders that the human mind can discover after all." (Sharon Begley, "New Answer for an Old Question", Newsweek, 5 July, 1993)
"At first glance the theory of numbers is deprived of any geometricity. But this is actually not the case. At the contemporary stage of development of computers it has become possible to explain to a wide range of readers that visual geometry helps not only to illustrate some abstract situations from the number theory, but sometimes also to solve new problems."
"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)
"Number theory is so difficult, albeit so fascinating, because mathematicians try to examine additive creations under a multiplicative light." (William Dunham, 1994)
"Yet, I believe the problem stands like a unconquerable fortress. For all that is known, it would be almost by luck that an odd perfect number would be found. On the other hand, nothing that has been proved is promising to show that odd perfect numbers do not exist. New ideas are required." (Paulo Ribenboim, "The New Book of Prime Number Records", 1996)
"We should not leave unmentioned the principal numbers […] those which are called ‘perfect numbers’. These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason twenty-eight, four hundred ninety-six, and eight thousand one hundred twenty-eight are called perfect numbers." (Stanley J Bezuszka & Margaret J Kenney, "Even perfect numbers", Math. Teacher 90, 1997)
"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)
"Suppose that we think of the integers lined up like dominoes. The inductive step tells us that they are close enough for each domino to knock over the next one, the base case tells us that the first domino falls over; the conclusion is that they all fall over. The fault in this analogy is that it takes time for each domino to fall and so a domino which is a long way along the line won't fall over fora long time. Mathematical implication is outside time." (Peter J Eccles, "An Introduction to Mathematical Reasoning", 1997)
"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)
"Sequences of random numbers also inevitably display certain regularities. […] The trouble is, just as no real die, coin, or roulette wheel is ever likely to be perfectly fair, no numerical recipe produces truly random numbers. The mere existence of a formula suggests some sort of predictability or pattern." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)
"The practical definitions of randomness - a sequence is random by virtue of how many and which statistical tests it satisfies and a sequence is random by virtue of the length of the algorithm necessary to describe it [...]." (Deborah J. Bennett, "Randomness", 1998)
"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)
"As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of arithmetic is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant." (Gerald Tenenbaum & Michael M France, "The Prime Numbers and Their Distribution", 2000)
"One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the ‘ideal gases’ which physicists are so fond of. Considered from an external point of view, the distribution is - in broad terms - deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted." (Gerald Tenenbaum & Michael M France, "The Prime Numbers and Their Distribution", 2000)
"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, "Uncle Petros and Goldbach's Conjecture", 2000)
"Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall and Carl Pomerance, "PrimeNumbers: A Computational Perspective", 2001)
"Throughout both ancient and modern history the feverish hunt for perfect numbers became a religion." (Clifford A Pickover, "Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning", 2001)
"Although the prime numbers are rigidly determined, they somehow feel like experimental data." Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)
"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, "The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics", 2002)
"The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)
"[…] despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge." (Marcus du Sautoy, "The Music of the Primes", 2003)
"Our world resonates with patterns. The waxing and waning of the moon. The changing of the seasons. The microscopic cell structure of all living things have patterns. Perhaps that explains our fascination with prime numbers which are uniquely without pattern. Prime numbers are among the most mysterious phenomena in mathematics." (Manindra Agrawal, 2003)
"The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle." (Marcus du Sautoy, "The Music of the Primes", 2003)
"The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes." (David Ruelle, "The Mathematician's Brain", 2007)
"Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from truly touching. […] If you go on counting, you discover that these pairs gradually become rarer, lost in that silent, measured space made only of ciphers. You develop a distressing presentiment that the pairs encountered up until that point were accidental, that solitude is the true destiny. Then, just when you’re about to surrender, you come across another pair of twins, clutching each other tightly." (Paolo Giordano, "The Solitude of prime numbers", 2008)
"[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime." (Martin Gardner, "When You Were a Tadpole and I Was a Fish", 2009)
"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)
"Number theory is so difficult, albeit so fascinating, because mathematicians try to examine additive creations under a multiplicative light." (William Dunham, 1994)
"Yet, I believe the problem stands like a unconquerable fortress. For all that is known, it would be almost by luck that an odd perfect number would be found. On the other hand, nothing that has been proved is promising to show that odd perfect numbers do not exist. New ideas are required." (Paulo Ribenboim, "The New Book of Prime Number Records", 1996)
"We should not leave unmentioned the principal numbers […] those which are called ‘perfect numbers’. These have parts which are neither larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason twenty-eight, four hundred ninety-six, and eight thousand one hundred twenty-eight are called perfect numbers." (Stanley J Bezuszka & Margaret J Kenney, "Even perfect numbers", Math. Teacher 90, 1997)
"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)
"Suppose that we think of the integers lined up like dominoes. The inductive step tells us that they are close enough for each domino to knock over the next one, the base case tells us that the first domino falls over; the conclusion is that they all fall over. The fault in this analogy is that it takes time for each domino to fall and so a domino which is a long way along the line won't fall over fora long time. Mathematical implication is outside time." (Peter J Eccles, "An Introduction to Mathematical Reasoning", 1997)
"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)
"Sequences of random numbers also inevitably display certain regularities. […] The trouble is, just as no real die, coin, or roulette wheel is ever likely to be perfectly fair, no numerical recipe produces truly random numbers. The mere existence of a formula suggests some sort of predictability or pattern." (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)
"The practical definitions of randomness - a sequence is random by virtue of how many and which statistical tests it satisfies and a sequence is random by virtue of the length of the algorithm necessary to describe it [...]." (Deborah J. Bennett, "Randomness", 1998)
"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)
"As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of arithmetic is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant." (Gerald Tenenbaum & Michael M France, "The Prime Numbers and Their Distribution", 2000)
"One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the ‘ideal gases’ which physicists are so fond of. Considered from an external point of view, the distribution is - in broad terms - deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted." (Gerald Tenenbaum & Michael M France, "The Prime Numbers and Their Distribution", 2000)
"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, "Uncle Petros and Goldbach's Conjecture", 2000)
"Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall and Carl Pomerance, "PrimeNumbers: A Computational Perspective", 2001)
"Throughout both ancient and modern history the feverish hunt for perfect numbers became a religion." (Clifford A Pickover, "Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning", 2001)
"Although the prime numbers are rigidly determined, they somehow feel like experimental data." Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)
"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, "The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics", 2002)
"The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers, "Mathematics: A Very Short Introduction", 2002)
"[…] despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge." (Marcus du Sautoy, "The Music of the Primes", 2003)
"Our world resonates with patterns. The waxing and waning of the moon. The changing of the seasons. The microscopic cell structure of all living things have patterns. Perhaps that explains our fascination with prime numbers which are uniquely without pattern. Prime numbers are among the most mysterious phenomena in mathematics." (Manindra Agrawal, 2003)
"The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle." (Marcus du Sautoy, "The Music of the Primes", 2003)
"The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes." (David Ruelle, "The Mathematician's Brain", 2007)
"Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from truly touching. […] If you go on counting, you discover that these pairs gradually become rarer, lost in that silent, measured space made only of ciphers. You develop a distressing presentiment that the pairs encountered up until that point were accidental, that solitude is the true destiny. Then, just when you’re about to surrender, you come across another pair of twins, clutching each other tightly." (Paolo Giordano, "The Solitude of prime numbers", 2008)
"[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime." (Martin Gardner, "When You Were a Tadpole and I Was a Fish", 2009)
"While number theory looks for patterns in sequences of numbers, dynamical systems actually produce sequences of numbers [...]. The two merge when mathematicians look for number-theoretic patterns hidden in those sequences." (Kelsey Houston-Edwards, "Mathematicians Set Numbers in Motion to Unlock Their Secrets", Quanta Magazine, 2021)
"We found a beautiful and most general proposition, namely, that every integer is either a square, or the sum of two, three or at most four squares. This theorem depends on some of the most recondite mysteries of numbers, and it is not possible to present its proof on the margin of this page." (Pierre de Fermat)
"A prime number, which exceeds a multiple of four by unity, is only once the hypotenuse of a right triangle." (Pierre de Fermat)
"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate." (Leonhard Euler)
"The existence of an odd perfect number – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle." (James J Sylvester)
"The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations." (Godfrey H Hardy)
"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)
"We found a beautiful and most general proposition, namely, that every integer is either a square, or the sum of two, three or at most four squares. This theorem depends on some of the most recondite mysteries of numbers, and it is not possible to present its proof on the margin of this page." (Pierre de Fermat)
"A prime number, which exceeds a multiple of four by unity, is only once the hypotenuse of a right triangle." (Pierre de Fermat)
"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate." (Leonhard Euler)
"The existence of an odd perfect number – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle." (James J Sylvester)
"The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations." (Godfrey H Hardy)
"One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery - and more totally useless - than the perfect numbers." (Martin Gardner)
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