Rediscovery of lost mathematics

Question

Archimedes (ca. 287-212BC) described what are now known as the 13 Archimedean solids in a lost work, later mentioned by Pappus. But it awaited Kepler (1619) for the 13 semiregular polyhedra to be reconstructed.
   
     ^{(Image from tess-elation.co.uk/johannes-kepler.)}
So there is a sense in which a piece of mathematics was "lost" for 1800 years before it was "rediscovered."

Q. I am interested to learn of other instances of mathematical results or insights that were known to at least one person, were essentially correct, but were lost (or never known to any but that one person), and only rediscovered later.

1800 years is surely extreme, but 50 or even 20 years is a long time in the progress of modern mathematics.

Because I am interested in how loss/rediscovery might shed light on the inevitability of mathematical ideas, I would say that Ramanujan's Lost Notebook does not speak to the same issue, as the rediscovery required locating his lost "notebook" and interpreting it, as opposed to independent rediscovery of his formulas.

Answer 1

Just today, I read in the July 2014 Bulletin of the American Math Society, in the Mathematical Perspectives piece by Gerald Alexanderson, that "Lorenzo Mascheroni ... in ... 1797, proved that any [straight-edge and compass] construction ... can be carried out by compass alone. And that is where the problem stood until 1928 when a student browsing in a rack of books in a Copenhagen bookshop found a small book by Georg Mohr, an obscure Danish mathematician. It was ... published in 1672. It contained a proof of what was then called Mascheroni's Theorem. The contents of this volume had remained totally unknown."

Answer 2

The algorithm now known as the FFT, described in Cooley and Tukey's landmark 1969 paper, was known to Gauss and appears among his unpublished works around 1805. I have also read that Archimedes' discovery of (at least parts of) integral calculus was found in a Byzantine manuscript whose pages had been "recycled" - the precious work of Archimedes washed (imperfectly, thankfully) from the pages to be refilled with some Greek clerical mumbo-jumbo. This would put Archimedes almost 1900 years ahead of Newton and Leibniz in that particular discovery.

Answer 3

In 1994 integration (!) was discovered by medical researchers. The article A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves has very big ipact (google scholar gives 299 citations).

Abstract

OBJECTIVE To develop a mathematical model for the determination of total areas under curves from various metabolic studies.

RESEARCH DESIGN AND METHODS In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method Gess than ±0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin.

Answer 4

Bernhard Bolzano .... ( interesting reading ) Much of his work was unpublished until much later (for reasons see the link), thus remaining largely unknown. For example, a theorem of Weierstrass is now known as the "Bolzano-Weierstrass theorem", acknowledging that Bolzano had proved it previously. He anticipated Cantor and Dedekind in work on doing calculus without infinitesimals. His example of a continuous nowhere-differentiable function is in a manuscript from 1830, but only published in 1930.

Answer 5

The Schröder-Hipparchus numbers give an interesting example:

According to a line in Plutarch's Table Talk, Hipparchus showed that the number of "affirmative compound propositions" that can be made from ten simple propositions is 103049 and that the number of negative compound propositions that can be made from ten simple propositions is 310952. This statement went unexplained until 1994, when David Hough, a graduate student at George Washington University, observed that there are 103049 ways of inserting parentheses into a sequence of ten items. A similar explanation can be provided for the other number: it is very close to the average of the tenth and eleventh Schröder–Hipparchus numbers, 310954, and counts bracketings of ten terms together with a negative particle

The problem of counting parenthesizations was introduced to modern mathematics by Schröder (1870).

If this interpretation is correct it's a non-trivial combinatorial problem that Hipparchus solved. Hipparchus must have known some combinatorial techniques that are not in any contemporary sources that survive today but which were rediscovered in the 19th century.

Answer 6

Eratosthenes calculated the radius and circumference of the Earth with accuracy $\approx 2\%$. Of course it is "only applied" mathematics, but extremely advanced for that time. The next time the same accuracy was achieved was only in the 19th century.

Almost at the same time Aristarchus of Samos calculated the sizes of the Sun and Moon, as well as their distances from the Earth in terms of the Earth's radius. His Heliocentrism was rejected until it was successfully revived by Copernicus, after which Johannes Kepler described planetary motions with greater accuracy, with Kepler's laws, and Isaac Newton gave a theoretical explanation based on laws of gravitational attraction and dynamics.

In both cases we have results which were "lost" for almost $2000$ years.

Answer 7

There is a nice, slightly speculative, example in Conway and Doyle’s Division by Three paper:

In 1926 Lindenbaum and Tarski announced, in an infamous paper that contained statements (without proof) of 144 theorems of set theory, that Lindenbaum had found a proof of division by three. Their failure to give any hint of a proof must have frustrated Sierpiński, for it appears that twenty years later he still did not know how to divide by three. Finally, in 1949, in a paper ‘dedicated to Professor Wacław Sierpiński in celebration of his forty years as teacher and scholar’, Tarski published a proof. In this paper, Tarski explained that unfortunately he couldn’t remember how Lindenbaum’s proof had gone, except that it involved an argument like the one Sierpiński had used in dividing by two, and another lemma, due to Tarski, which we will describe below. Instead of Lindenbaum’s proof, he gave another.

…

We tried and tried and tried to adapt [Sierpiński’s] method to the case of dividing by three, but we kept getting stuck at the same point in the argument. So finally we decided to look at Tarski’s paper, and we saw that the lemma Tarski said Lindenbaum had used was precisely what we needed to get past the point we were stuck on! So now we had a proof of division by three that combined an argument like that Sierpiński used in dividing by two with an appeal to Tarski’s lemma, and we figured we must have hit upon an argument very much like that of Lindenbaum’s. This is the solution we will describe here: Lindenbaum’s argument, after 62 years.

Answer 8

This example has quite a short duration compared to the others listed so far but I thought I might mention it anyway: Wolfgang Döblin achieved important results in stochastic calculus shortly before his death in 1940 during military service. His proof of Itō's formula (which would not be proved by Itō until 1944) was recorded in a sealed envelope which was not opened until 2000, and has led to that result's being renamed the Itō-Döblin theorem in some textbooks.

Answer 9

The fact that any continuous self map of an interval with a point of period three must have periodic points of all periods was proved in the paper "Period Three Implies chaos" by Li and Yorke which was published in 1975. This was considered new at the time but was later seen to be a very special case of Sharkovskii's theorem (which gives a complete ordering of the natural numbers such that a point of period $n$ implies points of all periods following it in this ordering). Sharkovskii had published his theorem a decade earlier in 1964.

Answer 10

The Rogers-Ramanujan identities have a similar story. They were discovered and proved by Leonard James Rogers in 1894 and then promptly forgotten. Ramanujan then discovered them in 1913 without a proof.

Answer 11

If "essentially correct" is not necessarily the same as "rigorously established", then Lucjan Emil Boettcher qualifies with his pioneering work in what is now known as holomorphic dynamics, 20 years prior to Pierre Fatou and Gaston Julia (without using the notion of normal family, though, which was not available when he was working on these topics), and writing in German, Polish and Russian. See Mathematicians whose works were criticized by contemporaries but became widely accepted later

Answer 12

There is René Descartes' formula relating the radii of four mutually touching circles, which he sent to Princess Elisabeth of the Palatinate in 1643, rediscovered in 1826 by Jakob Steiner, in 1842 by Philip Beecroft, and again in 1936 by chemistry Nobel prize winner Frederick Soddy who announced his discovery in Nature in verse:

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum. 

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

It is curious, given Soddy's "kissing", that Beecroft published his result in The Lady's and Gentleman's Diary.

Answer 13

Low-density parity-check codes (LDPC) were invented by Robert Gallager in his PhD thesis (1960). LDPC codes were forgotten until his work was rediscovered in 1996. Similar to Gallager’s LDPC codes were reinvented by different communities at roughly the same time.

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ADDENDUM: Gallager has the following comments to make:

LDPC was not quite forgotten; I wrote a monograph on it, published by MIT Press in 1963, and that monograph is still in print (although never a big seller). I investigated the possibilities for its use in a few military and space applications in the 60's and 70's as best I remember, but solid state technology was still too primitive for it to be economically feasible.

I feel quite remiss for not even mentioning it in a pretty substantial chapter on error correction coding in my Information Theory and Reliable Communication text book in 1968, nor in my Principles of Digital Communication text book in 2008. It was a deliberate choice rather than forgetfulness in both cases. In 1968, I didn't foresee the technology becoming better so fast, and in 2008 it had become a highly specialized technique that I didn't feel was the most important thing for first year graduate students to focus on.

When it was rediscovered, I was interested in following what others were doing on it, but was more interested in the other research topics I was working on at the time. At that time, what was needed was people really interested in detailed implementation, standardization, and manufacturing, and somehow I felt that others could do that better than I. There were enough people who were still familiar with my earlier work to focus again on it when it was reinvented, so the newer ideas and the older quickly became merged.

Technology works in strange ways, with some people more focused on basic research, some (particularly at places like Lincoln Labs, JPL, Qualcomm, Motorola) focusing more on using basic research, some motivated toward starting companies, and some in building things at well established companies and research labs. It has always amazed me that there are enough people at the interfaces that good ideas often become used when the time is ripe.

Answer 14

Dehn's proof that the mapping class group is finitely generated by twists, in 1938, was independently discovered (and simplified) by Lickorish in 1962 and 1964. Dehn's coordinates for curves on surfaces, also in 1938, was rediscovered by Thurston (and greatly extended) in 1988.

In all cases I've give the publication dates; the relevant mathematics was being publicized earlier: at least 1922 for Dehn and 1976 for Thurston. But Dehn's work was interrupted by two world wars... See Stillwell's remarks in his collected translations "Papers on group theory and topology".

Answer 15

Branko Grunbaum wrote a paper called "Lectures on lost mathematics" where he mentioned various mathematical results, theories and problems that have been lost.

Answer 16

The number theory work of Fermat might be an example. He was rather secretive about his methods and much has to be rediscovered later by Euler. This includes Fermat's two-square theorem: It was first mentioned by Fermat as a theorem in a 1640 letter to Mersenne and also analogous statements about primes numbers of the form $x^2+2y^2$ and $x^2+3y^2$ were made in a 1654 letter to Pascal. While Fermat claimed to have solid proofs, he did not write more than a very vague sketch using infinite descent. Euler first became aware of Fermat's work around 1730. It took Euler until 1749 to prove Fermat's two-square theorem and until 1772 to prove the analogous statements about primes numbers of the form $x^2+2y^2$ and $x^2+3y^2$.

A more knowledgeable person could certainly present more examples in the work of Fermat and Euler. I do not include "Fermat's last theorem" as it seems virtually impossible to me that Fermat possessed a correct proof for this.

Answer 17

There is also Redfield who discovered the cycle index series and anticipated combinatorial species in enumerative combinatorics. His first paper was published but ignored. His second paper was rejected for publication. The cycle index series was then rediscovered by Polya ten years later.

Redfield, J. Howard (1927). "The Theory of Group-Reduced Distributions". American Journal of Mathematics 49 (3): 433–455. doi:10.2307/2370675. JSTOR 2370675. MR 1506633.

G. Pólya (1937). "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen". Acta Mathematica 68 (1): 145–254. doi:10.1007/BF02546665.

Answer 18

The Cauchy-Davenport Theorem

Let $t$ be a non-negative integer and let $x_1, ..., x_t$ be nonzero elements of $\mathbb{Z}/p$ which are not necessarily distinct. Then the number of elements of $\mathbb{Z}/p$ that can be written as the sum of some subset (possibly empty) of the $x_i$ is at least $\min\{p,t+1\}$. In particular, if $t\geq p-1$, then every element of $\mathbb{Z}/p$ can written in this way.

Davenport proved this result in 1935, which is used quite extensively in the circle method and the Waring's problem, without knowing that in fact this was a result proved by Cauchy in 1813.

Davenport, H, A historical note. J. London Math. Soc. 22, (1947). 100–101

Answer 19

The Tonelli–Shanks algorithm for finding square roots modulo a prime number. Today this is well known, it is e.g. used in the quadratic sieve method to factor large numbers. Alberto Tonelli discovered the algorithm in 1891, it was re-discovered by Daniel Shanks in 1973.

Answer 20

In a series of 3 articles ($*$), Luigi Onofri studied the group of permutations $\mathrm{Sym}(I)$ of an infinite countable set $I$. He notably proved (in the third article, 1929) that this group has exactly 4 normal subgroups: $$\{1\}\subset\mathrm{Alt}(I)\subset\mathrm{Sym}_{\mathrm{fin}}(I)\subset\mathrm{Sym}(I)$$ where $\mathrm{Sym}_{\mathrm{fin}}(I)$ is the subgroup of finitely supported permutations ("substitutions operating on finitely many elements") and $\mathrm{Alt}(I)$ is its subgroup of index 2 of even finitely supported permutations.

See https://math.stackexchange.com/a/2645097/35400 for more details.

This now classical result was rediscovered by Schreier and Ulam (1933, ${*}{*}$) and is so far exclusively attributed to Schreier-Ulam [with the notable but isolated exception of (${*}{*}{*}_1$), 1956 and (${*}{*}{*}_2)$, 1962], often as Baer-Schreier-Ulam theorem, referring to a subsequent extension (1934) by Baer to a classification of normal subgroups of the symmetric group over an arbitrary infinite set.

($*$) L. Onofri. Teoria delle sostituzioni che operano su una infinità numerabile di elementi, Memorie I, II, III. Annali di Matematica Pura ed Applicata. Memoria I: vol. 4(1) 73-106, 1927; Memoria II: vol 5(1), 147-168, 1928; Memoria III: vol. 7(1), 103-130. (restricted Springerlink: Memoria I, Memoria II, Memoria III)

(${*}{*}$) J. Schreier, S. Ulam. Über die Permutationsgruppe der natürlichen Zahlenfolge. Studia Mathematica (1933) Vol. 4(1), p.134-141, 1933. (EUDML unrestricted access)

(${*}{*}{*}_1$) W. Scott. The infinite symmetric and alternating groups. Pages 1-22 in: W. Scott, C. Holmes, E. Walker, Contributions to the theory of groups, National Science Foundation Research Project on Geometry of Function Space, report no 5, NSF-G 1126, U. of Kansas, 1956. (Quite rare book, not on MathSciNet; I got it sent to my library from another one. Seems to have been reedited recently (Amazon Link); Publisher: Literary Licensing, LLC (March 30, 2013) ISBN-10: 1258647044; ISBN-13: 978-1258647049; Year 2013.)

(${*}{*}{*}_2$) C. Kent. Constructive analogues of the group of permutations of the natural numbers. Trans. Amer. Math. Soc. 104 1962 347–362. (unrestricted pdf access)

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Added: at the opposite, there is another famous result, originally due to the same authors Jósef Schreier and Stanislaw Ulam, for which they are scarcely quoted, namely: they proved that the group of orientation-preserving self-homeomorphisms of the circle is a simple group (${*}{*}{*}{*}$). (Still Google Scholar detects a certain number of quotations, so this would not qualify as "forgotten".)

(${*}{*}{*}{*}$) J. Schreier; S. Ulam. Eine Bemerkung über die Gruppe der topologisehen Abbildungen der Kreislinie auf sich selbst. Studia math. 5, 155-159 (1934). (EUDML unrestricted access)

Answer 21

Leibniz proved that $$1-\frac13+\frac15-\frac17+\frac19-\frac{1}{11}+\cdots = \frac{\pi}{4},$$ in 1674. However this identity was known 300 years before to the Kerala School (India) identified with Madhava (see e.g. Borwein 2014; pdf).

In 1976 E. Salamin and R. Brent obtained independently a formula for approximating the number $\pi$ which is quadratically convergent; that is, every term one adds doubles the number of correct digits of $\pi$. We now know that this formula had been already proved by Gauss. It appears on page 6 of handbook 6, "Short Essays from various fields of mathematics, begun in May 1809", but the formula was forgotten. One reason could be that it is not of practical use if one has to make the calculations by hand. For more information see the chapter 7 of the book "$\pi$ unleashed" by J. Arndt and C. Haenel (Springer).

Answer 22

The number of binary De Bruijn sequences of order $n$ was determined by Camille Flye Sainte-Marie in 1894, but the result was forgotten and rediscovered by De Bruijn in 1946. However, the practice of calling them De Bruijn sequences remains entrenched.

Answer 23

The breadth-first search, a classical algorithm of Computational Graph Theory, has an interesting history of invention.

According to Wikipedia, it was invented in 1945 by Michael Burke and Konrad Zuse, in his (rejected) Ph.D. thesis on the Plankalkül programming language, but this was not published until 1972. It was then reinvented in 1959 by E. F. Moore, who used it to find the shortest path out of a maze, and discovered independently by C. Y. Lee as a wire routing algorithm (published 1961).

Answer 24

The work of Charles Riquier (1853-1929) and Maurice Janet (1888-1983) on the formal properties of systems of partial differential equations (published 1910 - 1929) fell out of sight until J.-F. Pommaret modernised and promoted it (in his 1978 book Systems of Partial Differential Equations and Lie Pseudogroups).

Ernest Vessiot (1865 - 1952) is remembered today, if at all, for his contributions to differential Galois theory. In 1924 he produced a vector field formulation of partial differential equations (dual to Cartan's Pfaffian formulation) which was forgotten until 1985 (see paper by E. Fackerell in R. Martini Geometric Aspects of the Einstein Equations and Integrable Systems, Lecture Notes in Physics 239, Springer). In 1939 Vessiot used this formulation to link Darboux integrable second order PDEs to Lie groups of dimension $\leq 3$. This was subsequently extended by P. Vassiliou in 1986.

Answer 25

David S. Richeson, in his book, Tales of Impossibility, tells the story of Pierre Wantzel. I'll quote bits and pieces:

In 1837 ... Wantzel proved that it was impossible to trisect every angle, to construct every regular polygon, and to double the cube.... The result came with a deafening silence. Not only was it not publicized at the time, prominent mathematicians even a century later did not know who proved the impossibility results.

Wantzel published his article in one of the premier journals of the time [J. Math. Pures Appl.]. And yet his work was almost immediately forgotten.... On December 18, 1852, Sir William Rowan Hamilton wrote to De Morgan, "Are you sure that it is impossible to trisect the angle by Euclid?" De Morgan replied on Christmas Eve, "As to trisection of the angle, Gauss' discovery increases my disbelief in its possibility."

In 1897 Felix Klein wrote a book called Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle. In the introduction he wrote [The proof of the impossibility of the duplication of the cube and the trisection of an arbitrary angle] "is implicitly involved in the Galois theory as presented today in treatises on higher algebra." Klein did not mention Wantzel. Moreover, further muddying the water he incorrectly credited Gauss with the proof of the impossibility of constructing all regular polygons.

In 1914 Raymond Archibald ... wrote, "Who first proved the impossibility of the classic problem of trisection of an angle? I have not met with a statement of this fact in any of the mathematical histories...."

James Pierpont [1895] did squash the Gauss misinformation, but he did not give Wantzel credit.

Many mathematics books ... in the late nineteenth and early twentieth centuries discussed the classical problems but did not include their eventual solutions.... Often they misattributed the polygon proof to Gauss. As for the proofs of impossibility for angle trisection and the doubling of the cube – they either didn't know whether it had been proved, didn't know who gave the first proof, or misattributed the proof.

Repeatedly throughout the twentieth century – even as late as 1990 – mathematicians and historians of mathematics overlooked Wantzel and his contributions [a footnote refers to p. 152 of Eves' 1990 An Introduction to the History of Mathematics].

Answer 26

One of the most famous examples of "lost mathematics" must surely be the whole subject of Galois theory and, to some extent, group theory as developed by Galois. Galois attempted to publish his work several times, but it was overlooked by the likes of Cauchy and Fourier (in his defence, Fourier died). Luckily nine years after his death Galois's papers found their way to Liouville and they were published.

Under a strict interpretation of the question, Galois theory is perhaps not an example of "lost mathematics" because it not recreated by some one else. On the other hand, it definitely was "lost" and it was then later "rediscovered" by Liouville.

Answer 27

In the 1920's Ernst Steinitz discovered a remarkable theorem which today is known as Steinitz's Theorem - A graph G is isomorphic to the edge-vertex graph of a 3-dimensional convex polyhedron if and only if the graph is planar and 3-connected.

Only in the 1960's did the importance of what Steinitz had accomplished become clear when Branko Grünbaum and Theodore Motzkin recast/rediscovered what Steinitz had done in modern graph theory theory language that the importance of this remarkable result came to be exploited.

I would also like to comment on the phenomenon of "lost mathematics" in general. Perhaps another light in which to view the issue is that the mathematics at issue has gone to "sleep." Sometimes this occurs because the work is written in a language that does not have many readers. Sometimes the issue is that it was written by a person whose work does not have many "followers" and who did not have a broad context in which to understand what had been accomplished. Finally, often a "thread" of mathematics goes to sleep because with the mathematical tools of the time the line of work involved has gone as far as researchers at that time were able to carry it. When the "sleeping" thread gets reawakened and looked at, sometimes new ideas and tools are available to carry an earlier line of work much further.

Answer 28

Cusick T., Flahive M. The Markoff and Lagrange spectra. AMS, 1989, page 2:

The result that $\mu(\alpha) \ge \sqrt{5}$ for all real $\alpha$ is often referred to as Hurwitz's Theorem, even though it is contained in Markoff's work. This is because Hurwitz [1891] proved $\mu(\alpha) \ge \sqrt{5}$ directly, whereas the paper of Markoff [1879] approaches the problem via quadratic forms. In fact, even earlier, Korkine and Zolotareff [1873, pp. 369-370] stated the result $\sqrt{d(f)}/m(f)\ge \sqrt{5}$ and also stated that the next largest value of $\sqrt{d(f)}/m(f)$ is $\sqrt{2}$. Markoff [1879] refers to their work as the starting point for his own.

(The minimum $m(f)$ of an indefinite binary quadratic form $f(x,y) = ax^2 + bxy + cy^2$ with real coefficients and positive discriminant $d(f) = b^2 — 4ac$ is defined by $m(f) = \inf|f(x,y)|$, where the infimum is taken over all pairs of integers $x$, $y$ not both zero.)

Answer 29

If $A$ is an $n\times n$ Hermitian matrix with eigenvalues $\lambda_1(A), \ldots, \lambda_n(A)$ and $i,j=1,\ldots,n$, then the $j$th component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j), \ldots, \lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j$th row and column by the formula $$|v_{i,j}|^2 \prod_{k=1;k\ne i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1} (\lambda_i(A)-\lambda_k(M_j)).$$ The above fact has been rediscovered and forgotten multiple times; see Figure 1 of Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra, by Peter B. Denton, Stephen J. Parke, Terence Tao, and Xining Zhang.

Answer 30

Korselt's criterion (1899) for Carmichael numbers was born before Carmichael numbers (1910). Carmichael gave the same criterion in his article Note on a new number theory function.