Prime number

positive integer with exactly two divisors, 1 and itself

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

      Gaussian primes
with norm less than 500.


  • In the tenth book Euclid deals with certain irrational magnitudes; and since the Greeks possessed no symbolism for surds, he was forced to adopt a geometrical representation. Propositions 1 to 21 deal generally with incommensurable magnitudes. The rest of the book, namely, propostions 22 to 117, is devoted to the discussion of every possible variety of lines which can be represented by  , where a and b denote commensurable lines. There are twenty-five species of such lines, and that Euclid could detect and classify them all is in the opinion of so competent an authority as Nesselmann the most striking illustration of his genius. No further advance in the theory of incommensurable magnitudes was made until the subject was taken up by Leonardo and Cardan after the interval of more than a thousand years.
    In the last proposition of the tenth book [prop. 117] the side and diagonal of a square are proved to be incommensurable. The proof is so short and easy that I may quote it. If possible let the side be to the diagonal in a commensurable ratio, namely, that of two integers, a and b. Suppose this ratio reduced to its lowest terms so that a and b have no common divisor other than unity, that is, they are prime to one another. Then (by Euc. I, 47)  ; therefore   is an even number; therefore b is an even number; hence, since a is prime to b, a must be an odd number. Again, since it has been shown that b is an even number, b may be represented by 2n; therefore  ; therefore  ; therefore   is an even number; therefore a is an even number. Thus the same number a must be both odd and even, which is absurd; therefore the side and the diagonal are incommensurable. Hankel believes that this proof is due to Pythagoras, and this is not unlikely. This proposition is also proved in another way in Euc. X, 9, and for this and other reasons it is now usually believed to be an interpolation by some commentator on the Elements.
  • Fermat's Last Theorem is to the effect that no integral values of x, y, z can be found to satisfy the equation xn+yn=zn if n is an integer greater than 2. ...It is possible that Fermat made some... erroneous supposition, though it is perhaps more probable that he discovered a rigorous demonstration. At any rate he asserts definitely that he had a valid proof—demonstratio mirabilis sane—and the fact that no theorem on the subject which he stated he had proved has been subsequently shown to be false must weigh strongly in his favour; the more so because in making the one incorrect statement in his writings (namely, that about binary powers) he added that he could not obtain a satisfactory demonstration of it. … [I]t took more than a century before some of the simpler results which Fermat had enunciated were proved, and thus it is not surprising that a proof of the theorem which he succeeded in establishing only towards the close of his life should involve great difficulties. ...I venture however to add my private suspicion that continued fractions played a not unimportant part in his researches, and as strengthening this conjecture I may note that some of his more recondite results—such as the theorem that a prime of the form 4n+1 is expressible as the sum of two squares—may be established with comparative ease by properties of such fractions.
  • Prime numbers belong to the exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoys 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... vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes...
  • In... 1859 Bernhard Riemann... presented a paper to the [Berlin] Academy... "On the Number of Prime Numbers Less Than a Given Quanitity." ...Riemann tackled the problem with the most sophisticated mathematics of his time... inventing for his purposes a mathematical object of great power and subtlety. ...[H]e made a guess about that object, and then remarked:

    One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective...

    That ... guess lay almost unnoticed for decades. Then... gradually seized... imaginations... until it attained the status of an overwhelming obsession. ...The Riemann Hypothesis... remained an obsession all through the twentieth century and remains one today, having resisted every attempt at proof and disproof. [It is] now the great white whale of mathematical research.
    • John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)
  • A prime number is one (which is) measured by a unit alone.
    Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος.
    ...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) 2nd ed., Vol. 1 p. 117. [Editor's note:] This theorem is not general, for when the difference of the two numbers is 1, and their sum is a prime, it is evident that the difference of the two squares is also a prime: thus  , &c. In fact every prime number is the difference of two integral squares.
  • Leonhard Euler stated that 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 human mind would never penetrate.
  • Remington Hale declares he has the prime number equation, which determines whether a number is prime or not.
  • And this proposition is generally true for all progressions and for all prime numbers; the proof of which I would send to you, if I were not afraid to be too long.
    Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long.
  • 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. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, 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.
  • It has the very commendable aim of contributing towards stressing the cultural side of mathematics. ...there appears the widespread interchange of the definitions of excessive and defective numbers. is stated that Euclid contended that every perfect number is of the form 2n-1(2n -1). It is true that Euclid proved that such numbers are perfect whenever 2n - 1 is a prime number but there seems to be no evidence to support the statement that he contended that no other such numbers exist. is stated that the arithmetization of mathematics began with Weierstrass in the sixties of the last century. The fact that this movement is much older was recently emphasized by H. Wieleitner... it is stated that the arithmos of Diophantus and the res of Fibonacci meant whole numbers, and... we find the statement that in the pre-Vieta period they were committed to natural numbers as the exclusive field for all arithmetic operations. On the contrary, operations with common fractions appear on some of the most ancient mathematical records.
    • G. A. Miller, Review of Number: The Language of Science (1931) Bull. Amer. Math. Soc. Vol. 37, Number 1.
  • The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics. A striking example is that of the distribution of prime numbers. The solution of this problem lies in finding a general formula which tells us the number of primes that lie in any given numerical interval. ...Edmund Landau ...wrote two large volumes analyzing this problem without solving it, using the most advanced mathematics known at the time. Even in the elementary aspects of mathematics we are thus dealing with complex topics which make great demands on our mathematical skills.
    • Lloyd Motz & Jefferson Hane Weaver, Conquering Mathematics: From Arithmetic to Calculus (1991) Reference Handbuch Der Lehre Von Der Verteilung Der Primzahlen (1909) Vol. 1 & 2.
  • Although I firmly believe that there is no such thing as a stupid question, there can indeed be stupid answers. 42 is an example. Not only is this a poor ripoff of Doug Adams' Hitchhiker's Guide, but it isn't even a prime number. Everyone surely knows that numerical answers to profound questions are always prime. (The correct answer is 37.)
  • There is another lesser-known "quote" of Erdos that I know first-hand he did not say, since I made it up! ...
    I gave a talk about Erdos and number theory, and I tried to explain how marvelous the Erdos--Kac theorem is...
    [Overhead Slide] Einstein: "God does not play dice with the universe."
    I then said orally: "I would like to think that Erdos and Kac replied..."
    [Overhead Slide] Erdos and Kac: Maybe so, but something is going on with the primes.
    ...Somehow, the San Diego newspaper picked this up the next day, and attributed it as a real quote of Erdos.
    • Carl Pomerance, describing Joint Math Meetings (January, 1997) San Diego impromptu session of remembrance following the death of Erdos, in a Pomerance communication brought to the attention of Matthew R. Watkins by Gerry Myerson in February, 2008.
  • I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras... I have no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as "startling". ...Very recently I came across a tract published by you styled Orders of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible. I would request that you go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly...
    • Srinivasa Ramanujan, Letter to G. H. Hardy, (January 16, 1913), published in Ramanujan: Letters and Commentary American Mathematical Society (1995) History of Mathematics, Vol. 9.
  • The present work is inspired by Edmund Landau's famous book, Handbuch der Lehre von der Verteilung der Primzahlen, where he posed two extremal questions on cosine polynomials and deduced various estimates on the distribution of primes using known estimates of the extremal quantities. Although since then better theoretical results are available for the error term of the prime number formula, Landau's method is still the best in finding explicit bounds. In particular, Rosser and Schonfeld used the method in their work "Approximate formulas for some functions of prime numbers".
    • Szilárd Révész, "The Least Possible Value at Zero of Some Nonnegative Cosine Polynomials and Equivalent Duel Problems," Abstract, Journal of Fourier Analysis and Applications: Kahane Special Issue (1995) John J. Benedetto, ed.
  • I would teach the world how the Greeks proved, more than 2,000 years ago, that there are infinitely many prime numbers. In my mind, this discovery is the beginning of mathematics – when humankind realised that, by pure thought alone, it could prove eternal truths of the universe.
    Prime numbers are the indivisible numbers, numbers that can be divided only by themselves and one. They are the most important numbers in mathematics, because every number is built by multiplying prime numbers together – for example, 60 = 2 x 2 x 3 x 5. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of numbers.
  • We don't naturally have some magic formula to try and find these prime numbers. In fact trying to find where the next prime number is represents one of the biggest mysteries in the whole of mathemaics. ...The challenge if the primes is somehow of the ultimate puzzle in the whole of mathematics.

A History of Mathematics (1893)

by Florian Cajori, source.
  • The Elements contains thirteen books by Euclid, and two, of which it is supposed that Hypsicles and Damascius are the authors. ...The seventh, eighth, ninth books are on the theory of numbers, or on arithmetic. In the ninth book is found the proof to the theorem that the number of primes is infinite.
  • Eratosthenes, eleven years younger than Archimedes, was a native of Cyrene. He... measured the obliquity of the ecliptic and invented a device for finding prime numbers. ...In his old age he lost his eyesight, and on that account is said to have committed suicide by voluntary starvation.
  • Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula   a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example   4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation.

The Music of the Primes (2003)

: Why an Unsolved Problem in Mathematics Matters by Marcus du Sautoy
  • Riemann's insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. ...[I]n the strange mathematical world beyond Riemann's glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony... would explain why outwardly the primes look so chaotic. The metamorphosis... where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there.
  • Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians.
  • Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist.
  • The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'
  • Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built.
  • In the spring of 1997, Connes went to Princeton to explain his new ideas to the big guns: Bombieri, Selberg and Sarnak. Princeton was still the undisputed Mecca of the Riemann Hypothesis... Selberg had become godfather to the problem... a man who had spent half a century doing battle with the primes. Sarnak... [had] recently joined forces with Nick Katz... one of the undisputed masters of the mathematics developed by Weil and Grothendieck. Together they had proved that the strange statistics of random drums that we believe describe the zeros in Riemann's landscape are definitely present in the landscapes considered by Weil and Grothendieck. ...It was Katz who, some years before, had found the mistake in Wiles's first erroneous proof of Fermat's Last Theorem. And finally there was Bombieri... the undisputed master of the Riemann Hypothesis. He had earned his Fields Meda for the most significant result to date about the error between the true number of primes and Gauss's guess - a proof of... the 'Riemann Hypothesis on average'. ...Bombieri, like Katz, has a fine eye for detail.

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