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While prime numbers are central objects in mathematics it looks that they were ignored and forgotten for long periods of time. I am interested to get some facts and insights about this matter, in particular:

1) Were prime numbers studied in ancient times only by the ancient Greeks? At what periods were they studied by the ancient Greeks themselves?

2) Is it the case that people largely or even entirely lost their interest in the prime numbers for about fifteen centuries until Fermat? What are the facts of the matter and what are the reasons that may explain these facts.

(motivated by conversations with Ron Livne.)

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Isn't that "people largely or even entirely lost their interest in Mathematics for about fifteen centuries", after the end of the Hellenistic era? –  Pietro Majer Sep 10 '12 at 22:13
No no no , not at all, and also in the Hellenistic era itself people continued interest in mathematics, even in number theory, but lost interest in prime numbers. –  Gil Kalai Sep 10 '12 at 22:31
The Ishango bone is pretty old and curiously has some suspicious prime numbers on it. I'm adding this as a comment because of lack of reasons for considering it as relevant, but I could not resist. P. –  Pasten Sep 10 '12 at 23:44
I don't understand the question, or why the answers given so far are not what you want. In which respect does your question differ from others where "prime number" is replaced by just about any problem in mathematics covered in the Elements and other works of the Greeks? And as for 1), a quick glance at some history book will reveal that we know next to nothing about when the Greeks studied what - we have Euclid, Nicomachus, and Diophantus, everything else is extrapolation. –  Franz Lemmermeyer Oct 14 '12 at 16:22
Dear Franz The answers give some nice information about question 1 and I will welcome more information and details. I am specifically asking about prime numbers and I would like to know to what extent they are present in the works of Nicomachus, and Diophantus and about studying them in other cultures/times. Also I am curious about question 2: what can be the explanation for the lost of interest in prime numbers for many centuries. –  Gil Kalai Oct 14 '12 at 21:18
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7 Answers

up vote 12 down vote accepted

The Liber Abaci (1202) of Fibonacci contains a chapter on perfect numbers and Mersenne primes (of course Mersenne came much later, but possibly slightly before Fermat; he is born slightly before Fermat but is essentially a contemporary).

I do not know if there are any new results; but at least it seems he was interested in them.

I am not sure if this counts as interested in prime numbers, but it is certainly number theory and involves primes very directly: the Chinses Remainder Theorem developped from about 3rd to 13th century in China (no surprise here); but also in 6th and 7th century in India.

A non-example would be the Chinese Hypothesis that used to be believed to originate in ancient China but did not.

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+1 for debunking the so-called Chinese Hypothesis. –  Charles Sep 10 '12 at 22:32
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In response to question (1), an authoritative source is Peter Rudman in "How Mathematics Happened: The First 50,000 Years". Some revelant quotes:

On the Ishango bone (20,000 BCE):

The concept of division, which must precede the concept of prime number, probably did not evolve until after 10,000 BCE and the emergence of herder-farmer cultures. The concept of prime numbers was probably only really understood after about 500 BCE by Greek mathematicians.

On the Babylonian clay tablet Plimpton 322 (1800 BCE):

This clay table shows that Babylonian scribes understood Pythagorean triples and perhaps the Pythagorean theorem. It also hints at some understandig of number concepts: prime numbers, composite numbers, regular numbers, rational numbers, and reduced fractions.

On the Sieve of Eratosthenes (250 BCE):

Is easy to apply and to understand. Babylonian scribes could have invented it more than one thousand years earlier --- but they apparently did not. Its invention was only possible after Pythagoras (500 BCE) and Euclid (300 BCE) had made the study of properties of numbers a subject worthy of the attention of Greek philosophers.

In response to question number 2, as described by O'Connor & Robertson, see also the Wikipedia entry, Islamic mathematicians were the heirs of the Greeks throughout the Middle Ages, motivated in part by their interest in practical applications of geometry and number theory to architecture and decoration. (Similarly, the Islamic law of inheritance served as a drive for the development of algebra.)

The translation by Islamic scholars of the mathematical works of Greek mathematicians was the principal route of transmission of these texts to the Middle Ages. For example, Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912), while the Latin translation had to wait until Xylander (1575).

Some notable Islamic heroes of prime numbers:

As noted by Stopple, the 9th century astronomer Thabit ibn Qurra studied prime numbers of the form $3\cdot 2^n-1$ (now called Thabit numbers).

Ibn Al-Haytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form $2^{k-1}(2^k - 1)$ where $2^k - 1$ is prime. As noted by John Stillwell, Al-Haytham is also the first person that we know to state the theorem that if $p$ is prime then $1+(p-1)!$ is divisible by $p$ (only proven 750 years later by Lagrange).

Al-Farisi (born 1260) stated and attempted to prove the fundamental theorem of arithmetic, on the unique factorization of an integer into prime numbers.

Finally, the "why" question: There are no comparable heroes in Mediaeval Europe. My surmise is that this is because Christianity, with its figurative art, did not stimulate the interest in geometric and numerical patterns to the same extent as Islam did.

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For 2), it depends a little on how you interpret the question. Primes in the abstract are covered in Chapter XVIII of Dickson's History of the Theory of Numbers, vol I. There's not much between Euclid and Euler.

On the other hand, primes of special forms related to perfect numbers or amicable pairs were written about extensively in the 15 centuries before Fermat. Admittedly, often incorrectly or with little content. In Chapter I of Dickson, Carolus Bovillus (1470-1553) claims that $2^n-1$ is prime if $n$ is odd, giving the example $511=2^9-1$. (In fact $7|511$). But it was not all nonsense. For example, Thabit ibn Qurra (836-901) showed that if $$ p=3\cdot 2^{k-1}-1, q=3\cdot 2^k-1, r=9\cdot 2^{2k-1}-1 $$ are all primes, then $$ m=p\cdot q\cdot 2^k, n=r\cdot 2^k $$ form an amicable pair: $s(m)=n$ and $s(n)=m$, where $s(k)$ is the sum of the proper divisors of $k$.

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The error with 511 is slightly shocking. –  quid Sep 10 '12 at 22:50
Not as shocking as the Grothendieck 57 (or 27) story... Not nearly as shocking, in fact, considering the dates (we've learned so early that $m|n \Rightarrow 2^m-1 | 2^n-1$ that $2^3-1 | 2^9-1$ feels obvious to us, but that's anachronistic). –  Noam D. Elkies Oct 15 '12 at 0:55
The Grothendieck story is so shocking that it sounds apocryphal. I don't think the shock of the error here is related to the approach you've cited; rather, I would have thought that anyone working with primes (even back then?) would note 511 is not even and not divisible by 3 or 5. So any real attempt to check divisibility would start at 7... –  Benjamin Dickman Oct 16 '12 at 5:26
As far as I know, in the Grothendieck story, the number 57 was a spontaneous reaction -- Grothendieck wouldn't have made this mistake in an article. –  Lennart Meier Oct 19 '12 at 11:45
My 'shock' regarding the 511 was due to reason Benjamin Dickman gives. Regarding Gr. I am with Lennart Meier. I never found this story (as I read/understood it) shocking at all, not even that surprising. My understanding of the story: He was sort-of pressured by somebody (in a conversation) to discuss something with an explicit example of a prime, instead of in general. He found this misguided/annoying . And thus responded something like: well, whatever, so take fifty-seven. (So while he likely intended to say a prime, there was no relevance to it at all.) –  quid Oct 19 '12 at 12:12
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In recent times it has been claimed that Bhaskara I (around 700) and more definitely Ibn al-Haytham (965 - 1040) were aware of Wilson's theorem. This is much earlier than Wilson's theorem was previously supposed to be known, so perhaps there is more to be discovered about early work on prime numbers.

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Who has claimed this? And based on what text of Bhaskara I? –  Marty Sep 10 '12 at 23:46
@Marty: for Bhaskara I the only source I've seen is Wikipedia. However, for al-Haytham there is more scholarly support: Rashed, Roshdi Ibn al-Haytham et le théorème de Wilson. Arch. Hist. Exact Sci. 22 (1980), no. 4, 305–321. –  John Stillwell Sep 11 '12 at 0:18
Thanks for the reference - I'll check it out. And I'll look through Bhaskara I for clues about primes as well. –  Marty Sep 11 '12 at 0:44
Ibn al-Haytham: "This being shown we say that this is a necessary property for any prime number, that is to say that for any prime number - which is the number that is a multiple only of the unit - if you multiply the numbers that precede each other in the way that we have introduced, and if we add one to the product, and if we divide the sum by each of the numbers before the prime number, there is one, and if it is divided by the prime number, nothing is left." –  Marty Sep 11 '12 at 7:19
Regarding Bhaskara I, I am just about ready to call it bunk. It is all over the internet, so it is popular bunk. I think that the bunk comes from misinterpreting the following: In Vol. 2, Ch II, p.59 of his "History of the Theory of Numbers", Dickson explains how both "Ibn al-Haitam (about 1000)" and "Bhascara (born, 1114 A.D.)" treated similar problems about finding a number which has given remainders when divided by 2,3,4,5,6. From the date, this would appear to be Bhaskara II. Al-Haitam's problem leads him to "Wilson's Theorem" context, but Wilson's theorem is absent from Bhaskara II. –  Marty Sep 11 '12 at 7:26
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According to the book of David Wells on prime numbers (see page 43), critics think that Diophantus (b. between A.D. 200 and 214, d. between 284 and 298 at age 84) knew (empirically, presumably) that every prime of the form $4n+1$ is a sum of two squares.

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Dear Richard, That's interesting.Other than that isnt it true that while being number theorist, primes are not present in Diophantus' work? –  Gil Kalai Oct 14 '12 at 21:11
Dear Gil, From what I read on the internet, it seems to be true that primes are not present in Diophantus' work, but I am not an expert. –  Richard Stanley Oct 14 '12 at 23:42
It's true that Diophantus does not mention the concept of prime number, but he seldom mentioned any general concept, and was content to illustrate general ideas by examples. This was enough for Fermat, who became interested in primes of the form $x^2+y^2$ after reading the remark (in Diophantus Book III, Problem 19) that 65 is a sum of two squares "due to the fact that 65 is the product of 13 and 5, each of which is the sum of two squares." –  John Stillwell Oct 15 '12 at 22:05
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Gil Kalai writes:

2) Is it the case that people largely or even entirely lost their interest in the prime numbers for about fifteen centuries until Fermat? What are the facts of the matter and what are the reasons that may explain these facts.

It depends on who the people here are!

(a) In the Arabic-speaking world, where mathematics was alive and well, prime numbers did not lose their interest; in fact, as John Stillwell said above, the statement "Wilson's theorem" dates from that period.

(b) In most of Europe, there was essentially no pure mathematics of interest throughout the Middle Ages. (About the one exception is Fibonacci, who of course got at least part of his mathematical education outside Europe.)

Still, it would not surprise me if prime numbers turned out to be one of the few things in what we call number theory that was ever discussed in Western Europe during the Middle Ages. Reason: the popularity of Nicomachus's Arithmetic, translated (freely) by Boethius.

Boethius' Latin version was destined to exert a great influence on subsequent encyclopedic authors of the sixth and seventh centuries and throughout the Middle Ages up to the sixteenth century. From the sixth to the twelth century, when Greek geometry had almost vanished and science was at its lowest ebb, Boethius's Arithmetic, for all its faults, preserved the ideal of a theoretical science. Not until the thirteenth century, when Jordanus de Nemore's Arithmetic appeared in ten books, do we have a theoretical arithmetic on the Euclidean model, complete with proofs.

E. Grant, A source book in medieval science, Harvard U Press, 1974.

From a quick look at Nicomachus's original, it seems to be almost entirely about properties of integers, which are sometimes given a mystical or moral significance. Primality appears as one noteworthy property among several, side by side with being odd, even, triangular, pentagonal, heptagonal, perfect, superparticular, heteromecic, etc.

(Nothing or almost nothing non-trivial seems to be shown about any of these.)

As for Diophantus's Arithmetic, (a) it could not have an influence in Western Europe during the Middle Ages, as it was unknown there, (b) at any rate, it is largely about what we now would call the (highly ingenious!) construction of rational maps from n-dimensional affine space to varieties. There's very little in Diophantus about integers, and that as auxiliary material. Hence the fact that he does not really discuss prime numbers as such does not tell us much.

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Bhaskaracharya in his Lilavati ( a compendium of math puzzles for his daugther) has several examples that include prime numbers

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