The old Greek did not consider 1 a number, so it was not a prime. The theorem of unique prime factorization excludes 1 to be a prime number. But in between probably at Euler's and Goldbach's times? Who can determine most precisely (probably by original papers) when 1 first became a prime number and when 1 had been called a prime number for the last time?
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24$\begingroup$ The real question is: when will $-1$ begin its career as a prime number? $\endgroup$– Kevin O'BryantCommented Jul 6, 2010 at 12:27
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6$\begingroup$ Why does this matter at all? $\endgroup$– Kevin H. LinCommented Jul 6, 2010 at 19:14
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7$\begingroup$ Nice question. Modern people are so sure about the fact that "1 is an integer number" and that "1 is not a prime number", that they look with that certain air of superiority to the people of the past centuries who ignored such elementary and obvious facts... forgetting that they were the people who actually invented these concepts and gave to us; and also forgetting that in any case 1 being prime is just a matter of definition and conventions (certainly, the best choice for many reason, at the moment). $\endgroup$– Pietro MajerCommented Jul 6, 2010 at 20:15
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6$\begingroup$ Kevin, I think Conway has for many years been referring to the "prime" -1 in formulae involving quadratic symbols. $\endgroup$– T..Commented Jul 7, 2010 at 4:52
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4$\begingroup$ Even though the ancients did not have the field theory to prove the impossibility of trisecting an angle, they understood about the field with one element. Remarkable! $\endgroup$– Tom GoodwillieCommented Apr 26, 2012 at 11:09
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5 Answers
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Both Euler and Goldbach counted 1 as a prime in certain situations (variants of Goldbach's conjecture), and did exclude 1 whenever it suited them (arithmetical functions). The question whether 1 is prime or not was not so terribly important before unique factorization was discovered as a fundamental principle by Gauss.
Edit. Here's a nice episode: when Wallis "solved" Fermat's challenge to find a cube whose sum of divisors is a square (Fermat had given the example $\sigma(343) = 20^2$) by claiming that $1$ does it, Fermat was insulted. Wallis then complained that Fermat was not content with this solution and pretended that this was because "some do not admit that $1$ is a number," and remarked that others do.
By the way, in his own solution of the problem $1+p+p^2+p^3 = x^2$ for primes $p$, Fermat showed that the only solutions are $p = 1$ and $p = 7$, so he counted $1$ as a prime number.
answered Jul 6, 2010 at 12:01
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4$\begingroup$ To echo that: when I was in elementary school, I was taught that a prime number is one that cannot be evenly divided by a number other than one and itself. And thus 1 is prime. When I was in middle school, I was taught greatest common divisors and least common multiples and how to compute them from prime factorizations. And that a number can always be written, in exactly one way, as a product of primes. And oh, by the way, 1 is not a prime. $\endgroup$ Commented Jul 6, 2010 at 12:35
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$\begingroup$ What situation exactly are you thinking of where you say Euclid counted 1 as prime? $\endgroup$– OliverCommented Jul 6, 2010 at 18:44
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$\begingroup$ @Oliver: I meant Goldbach, of course -( $\endgroup$ Commented Jul 6, 2010 at 21:40
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$\begingroup$ Ah, that makes rather more sense. $\endgroup$– OliverCommented Jul 11, 2010 at 20:35
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$\begingroup$ Can you elaborate a bit on what you mean by Gauss discovering unique factorization as a fundamental principle? Clearly it was known to Euler (see e.g. the Euler product formula for what later became known as Riemann's zeta function), and some people speculate that Fermat tricked himself into believing to have proved his last theorem by falsely assuming unique factorization to hold in arbitrary number rings (i.e. considering it a principle too fundamental). Do you mean they never consciously put the principle into words? This is not meant to criticize, I am seriously interested in the answer $\endgroup$– VincentCommented Aug 1, 2014 at 14:50
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My naive understanding is like this: most, but not all, of the ancient Greeks excluded one from the category of numbers ($\alpha\rho\iota\theta\mu o\varsigma$), hence from the primes (others excluded two as well). Speussippus (c.350BC) is a rare exception:
Speussippus, then, is exceptional among pre-Hellenistic thinkers in that he considers one to be the first prime number. [L. Taran, Speusippus of Athens: a critical study with a collection of the related texts and commentary, Philosophia antiqua, vol. 39-40, E.J. Brill, 1981]
This view held (mostly) until Stevin's argument that one was a number and his development of the reals (late 16th century)
In general, mathematics before Stevin is of one character and, after him, it is of another re ecting his contributions. In this regard, he is like Euclid: he stood at a watershed in the history of mathematics. And as with Euclid, he was so successful that, from our present day vantage point, it is hard to see the other side of that watershed. Over there, one was not a number; here and now, it is; even is a number, and i, and aleph nul. [C. J. Jones, The concept of one as a number, Ph.D. thesis, University of Toronto, 1978.]
Now we enter a period of confusion. The view of one as a non-number slowly begins to die out, and some (e.g., Brancker + Pell's table 1688) begin to list one as a prime.
Not a prime for Schooten 1657, Clerke 1682, Chales 1690, Ozanam 1691, Brunot 1723, Cortes 1724, Reyneau 1739, Euler 1770, Horsley 1772, ... A prime for Wallis 1685, Goldbach 1742, Kruger 1746, Willich 1759, Lambert 1770, Felkel 1776, Warring 1782, ... (Not these folks may have used both views at times, as many of us alternate between ln and log for the natural log, depending on our audience)
The beginning of the end comes with Gauss' Disquisitiones Arithmeticae as the Fundamental theorem of arithmetic, and especially the uniqueness of factorization, becomes central. At about the same time number fields are introduced and the role of units becomes understood. I could give a long list (like the above) of yeses and nos in this period as well. But the choice of excluding one from the primes gains superiority and is now essentially universal among mathematicians.
Was there a time one was almost universally considered a prime? Absolutely not. Was there a time it was almost universally considered a non-prime: yes, much of history.
Edited to add "last time" from comments:
It might be that the last major mathematician "wrote" that one was a prime in print was G. H. Hardy, who lists one as a prime in his "A course of pure mathematics, "3rd ed., Cambridge University Press, 1921:
If there are only a finite number of primes let them be 1, 2, 3, 5, 7, 11, ... $N$. [section 61, page 143-144]
He does is indirectly later in this text:
the decimal $.111\ 010\ 100\ 010\ 10\ldots$, in which the $n$th figure is $1$ if $n$ is prime, and zero otherwise, represents an irrational number [section 78, page 174]
By the 9th edition, 1944, Gerry Myerson notes the first reference to 1 as prime is removed (I'd bet it was changed by the 7th edition, 1938, and will try to check). The decimal (surely accidentally) was still present in the 10th edition that I checked.
However, I am not convince Hardy personally thought (defined) that 1 was prime in 1921, I suspect he thought it was still not important enough to bother changing what he had written in the first edition, 1908. His comments in the article "The Theory of Numbers" in Science (New Series, Vol. 56, No. 1450, Oct. 13, 1922, pp. 401-405), appears to imply 1 is not prime. E.g., he repeats that Mersenne listed $2^n-1$ was prime for 2, 3, ... without commenting about $1=2^1-1$ or altering Mersenne's statement to start at 1---which was commonly done. (I admit this is this is weak evidence!)
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$\begingroup$ Very good...but...when was the last time some prominent mathematician held 1 to be a prime? $\endgroup$ Commented Apr 27, 2012 at 12:42
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1$\begingroup$ The question was "when 1 first became a prime number and when 1 had been called a prime number for the last time?" I am offering Speussippus (c.350BC) as the answer to the first part. As for "prominent mathematician" I have not seen one after Hardy's A course of pure mathematics, 3rd ed., Cambridge University Press, 1921 (pages 143-3, and page 174); unless you except what I think is an editing mistake. Recent edition of this text still have: The decimal .11101010001010... in which the nth figure is 1 if n is prime and zero otherwise, represents an irrational number. $\endgroup$ Commented Apr 27, 2012 at 13:35
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$\begingroup$ Welcome to MathOverflow! Gerhard "Ask Me About System Design" Paseman, 2012.04.27 $\endgroup$ Commented Apr 28, 2012 at 6:45
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$\begingroup$ @Chris: A very good answer that I would accept if I had access to my account. But unfortunately a computer crash a year ago makes that impossible. Now it only remains to pinpoint exactly when Hardy changed his opinion. Is there possibly any hint in a preface or else? Hans. $\endgroup$– user7116Commented Apr 28, 2012 at 7:00
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$\begingroup$ It would be nice to know when this changed in print, but, if the WorldCat database can believed, no library in the world has the 4th, 5th or 6th edition. The 7th, 1938, was a major revision, and is available in libraries at least Spain, Denmark, Germany and Slovenia. $\endgroup$ Commented Apr 28, 2012 at 16:16
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Wikipedia has lots of information on this topic. For example, "Henri Lebesgue is said to be the last professional mathematician to call 1 prime."
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1$\begingroup$ Wikipedia also notes that D N Lehmer (father of D H Lehmer) included 1 in his list of primes. I didn't know about Lebesgue, but I'd be inclined to discount him since he wasn't a number theorist so it didn't really matter whether he called 1 a prime or not. $\endgroup$ Commented Jul 6, 2010 at 12:03
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4$\begingroup$ @Gerry: that seems a bit territorial. Lebesgue was obviously a serious mathematician. I would prefer to give him the benefit of the doubt that he knew some number theory and had reasons (though I don't know what they would be) for his convention. $\endgroup$ Commented Jul 6, 2010 at 15:35
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4$\begingroup$ Is anyone here able to verify the claim concerning Lebesgue? Reference? I do not really trust the web in this case . . . $\endgroup$ Commented Jul 7, 2010 at 7:49
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3$\begingroup$ Franz, if you look at the wikipedia piece, it has a footnote directing you to page 33 of a recentish book by Derbyshire - have you checked that? I wonder if it's possible that someone was confusing Henri with another Lebesgue, a mid-19th century number theorist. $\endgroup$ Commented Sep 2, 2010 at 13:05
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$\begingroup$ Incidentally, David Williams has an easily findable article called "Brownian Motion and the Riemann Zeta-Function" in which he mentions how the proof of a probability conjecture (that a certain random variable is "mean zero ferromagnetic") would imply the Riemann hypothesis. (According to Chuck Newman, who David Williams cites, this implication was already known to Polya and can be found on page 424 of Polya's Collected Papers, Vol II.) So I'd be a bit cautious who you decide to ignore. $\endgroup$ Commented Sep 2, 2010 at 14:54
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Not a historical answer, but...
Ruminations on the "field with one element" are in a sense including 1 (and powers of 1) in the set of prime powers.
Also, in quantum groups (loosely speaking, one-parameter deformations of groups) the $q=1$ limit recovers the group while $q$ a root of unity is related to phenomena in prime characteristic $p$ > 0.
There is also the Krasner - Kazhdan - Deligne philosophy of "fields of characteristic $p$ as limits of fields of characteristic 0", recently and somewhat speculatively related to the field of one element in arxiv papers of Connes and Consani.
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2$\begingroup$ Then again, $1=2^0$, so it is a power of a prime even if it is not prime itself. $\endgroup$ Commented Jul 7, 2010 at 23:03
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1$\begingroup$ The putative field of one element, and its even more hypothetical "algebraic extensions", are meant as $p=1$ analogues of fields of size $p^1$ and $p^n$ respectively. Setting $n=0$ is closer to Krasner's approach (which has different aims and was not introduced as a theory of $F_1$). In the existing proposals for how to define these objects, $F_1$ and $F_{1^n}$ are different, and non-isomorphic, objects. $\endgroup$– T..Commented Jul 8, 2010 at 6:48
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1 is prime by Hardy's Theorem 90 (not Hilbert's!), see
http://groups.google.com/group/sci.math/msg/302ff4d9b99f2981
http://google.com/groups?selm=y8zoh20mtvm.fsf%5f-%[email protected]
answered Jul 7, 2010 at 18:06