real face of jnos bolyai - american mathematical society

Real Face of János BolyaiTamás Dénes

On the 150th anniversary of the death of János Bolyai1

In Vol. 56, No. 11, of Notices, I saw a fascinat-ing article with the title: Changing Faces—TheMistaken Portrait of Legendre. The author, Pe-ter Duren, writes with understandable bewil-derment: “It seems incredible that such egre-

gious error could have gone undetected for so manyyears.” This encouraged me to publish, in the samejournal, the story—no less fascinating—of the realface of János Bolyai.

It should be explained that in the case ofJános Bolyai two different interpretations of theword “face” are justified: the “face” in terms ofthe portrait (painting, drawing), and the “face” asan abstract concept. The first part of the articleintroduces the surprising story of his only portrait,which, although universally accepted, turns out notto be of him at all. The second part explores hisintellectual or “mind-face” (it is my own wordformation) and outlines a new approach to Bolyai’screative life and work.

The town of Marosvásárhely—which lies in theheartof theCentral EuropeanTransylvania—fulfillsan important role in the history of the Bolyai fam-ily: János lived most of his life there, and the locallibrary holds most of his manuscripts. The readermight find it strange that while János Bolyai is wellknown around the world as an eminent Hungar-ian, Transylvania (and so Marosvásárhely) can befound within the borders of Romania. This can beexplained by the unsettled history of Transylvania.If we look back only to the nineteenth century, thepart of Transylvania that was populated mainlyby Hungarians was autonomous at times, whereasat other times it belonged to Hungary. In 1947,following World War II, the Treaty of Paris gavethis area to Romania, so that is where it is foundon today’s maps.

Tamás Dénes is a mathematician and cryptologist fromHungary. His email address is [email protected] to the memory of Professor Elemér Kiss(1929–2006).

The article entitled “The real face of JánosBolyai” originates from my conversations withProfessor Elemér Kiss. Unfortunately his seriousillness, followed by his death in 2006, thwartedour writing of a shared article. This piece of workis intended to fill this gap.

Only Two Pictures of János Bolyai EverExisted, Neither of Which Has SurvivedJános Bolyai (December 15, 1802–January 29, 1860)emerges like a comet from the history of Hungarianmathematics.

“He was an illustrious mathematician with agreat mind; the first amongst the first”—read therecord of his death in the book of the ReformChurch of Marosvásárhely in Transylvania. OnNovember 3, 1823, he had sent a letter to his fatherfrom Temesvar, including the words that wouldlater become famous: “I created a new, differentworld out of nothing.”

What he meant by this “new world” was theidea of hyperbolic geometry, which was outlinedin 1832, as an appendix to the book Tentamenby Farkas Bolyai,2 entitled “The absolute true sci-ence of space” (Scientiam Spatii absolute veramexhibens). This, under the name “Appendix”, hasbecome his best known piece of writing.

The theory outlined in this work has beennamed “Bolyai-Lobachevsky geometry”, followinga decision made in 1894 at the International Bib-liographic Congress of Mathematical Sciences. InJanuary 2009 the “Appendix” by János Bolyai wasadded to UNESCO’s Memory of the World Register.

János Bolyai was the son of Farkas Bolyai—himself a defining figure of nineteenth-centuryHungarian mathematics, who was in regular corre-spondence with Gauss. As a result, it is perhapsnot surprising that Farkas Bolyai and his wifeZsuzsanna Árkosi Benkö were immortalized by

2Father of János Bolyai.

January 2011 Notices of the AMS 41

(Sou

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Figure 1. Title page of the “Appendix” by JánosBolyai.

contemporary artists in both drawings and oilpaintings. It might therefore be assumed thattheir child, who had already become famous inhis lifetime, would be immortalized in a similarmanner.

But based on contemporary sources, only twopictures of János Bolyai ever existed, neither ofwhich has survived. One of the “Vienna pictures”was mentioned by Farkas Bolyai himself in a letterwritten to his son on September 3, 1821. Accordingto other sources, by 1837 this picture could nolonger be found.

The other one was made while he was serving asa lieutenant. The destruction of this was accountedfor by János Bolyai himself: “I tore up this picture,which had been taken in a military parade, for Iwas not worthy of my father. I wasn’t craving theoutward immortality so wildly promoted by others.”

The most recent research on Bolyai, by profes-sors Tibor Weszely3 and Elemér Kiss,4 also supportsthe idea that there is no surviving authentic imageof János Bolyai.

3Sapientia University, Tg-Mures, Romania.4He had been a professor of mathematics until his death(2006) at the Sapientia University, Tg-Mures, Romania.

The Portrait of János Bolyai That Isn’tThe question is: how has this—supposedlyauthentic—portrait spread around the world withthe name of János Bolyai?

Well, exactly fifty years ago, on the centenaryof János Bolyai’s death, Hungarian and Romanianstamps were published with Bolyai’s name on them.Since then there has been an increased presence ofthis portrait everywhere: in books, on postcards,and most recently on the Internet, too. Today weknow for certain that this portrait is not that ofJános Bolyai.

Figure 2. The portrait, not of János Bolyai (onHungarian and Romanian stamps in 1960),that has been circulated around the world.

The year 2010 is the 150th jubilee of JánosBolyai’s death, so it’s about time that—after a latentperiod of fifty years—we resolve this scandalousmystery and bring to the public the results ofthe latest Bolyai research. In order to do this, Ifirst need to briefly introduce the reader to twocontemporaries of Bolyai: two Hungarian paintersand a painting that plays a key role in the story.

Mór Adler (1826–1902) was one of the pioneersof Hungarian painting. He stood out as a student ofsome merit at the Weisenberger School of GraphicArt, from which he went to the Vienna Academy.There he was taught by the then well-known histor-ical and religious painters, between 1842 and 1845.He then traveled to Munich in 1845 to study theworks of Zimneirmann and Schnorr von Carolsfeldand for further studies in 1846–1847 in Paris. Nexthe settled in Pest in 1848, where he would becomea respected figure in the art world by the end ofhis career. He took part in the Pest Artists Groupexhibition in 1851 and would take part in thisannually for the next fifty-eight years. He was bestknown for his portraiture and still-life paintings,which he executed in a fine realistic manner.

Mór Adler created the large oil painting above(150 x 100cm) in 1864.

The name of the person depicted in this paint-ing does not appear either on the front or theback of the painting, nor is it mentioned in anycontemporary documents. One thing we know for

42 Notices of the AMS Volume 58, Number 1

Figure 3. Oil painting by Mór Adler from 1864.

certain is that a drawing by Károly Lühnsdorf(1893–1958)5 was made, based on this painting.He wrote the name János Bolyai at the bottom ofthe drawing, accompanied by the following note:“I have drawn this portrait based on the only re-maining picture of János Bolyai, painted from life byMór Adler (1826–1902) artist from Óbuda in 1864—Károly Lühnsdorf .” The original drawing by KárolyLühnsdorf is now owned by the Bolyai family, butthe photo of it and Mór Adler’s painting can befound on the walls of the János Bolyai MathematicalSociety.

To sum up, taking account of Mór Adler’s andJános Bolyai’s biographical data and the fact thatthe painting is of a twenty-year-old man, we candraw the following conclusion. If the painting wasof János Bolyai, it would have to have been createdaround 1822, when Mór Adler wasn’t yet born.

Lühnsdorf states that he drew his picture “basedon Adler’s original painting from life”, a clearassumption that Adler painted his picture of Bolyaihimself. However, we know from biographical datathat Mór Adler traveled around Europe until 1848.Only then did he settle in Hungary—at which timeBolyai was already forty-six years old.

To assume that the painter did not paint fromlife but from memory would also be a mistake,as in 1826, when Mór Adler was born, Bolyai was

5He studied at the Hungarian Academy of Arts between1921 and 1928. His main interest was in painting por-traits and biblical scenes; he acquired fame in the fieldof portrait painting. These depicted scientists, historicalfigures, and personalities from religious and public life.

Figure 4. Károly Lühnsdorf’s drawing of MórAdler’s painting. On it, his handwritten notethat has, until now, misled the world.

already twenty-four years old. If they met towardthe end of the 1840s, when Adler was beginninghis artistic career, János Bolyai would have alreadybeen past forty.

Thus Mór Adler’s painting cannot be of JánosBolyai, and Károly Lühnsdorf must have written hisnote based on false information, thereby mislead-ing future generations. This is how this portrait,that IS NOT OF JÁNOS BOLYAI, started its journeyaround the world, being mistakenly recognized bymathematicians, students, and institutions as theonly original portrait of him.

The Real Face of János Bolyai“He was the first Hungarian mathe-matician who (according to LorándEötvös)6 created something world

6Loránd Eötvös (1848–1919) is remembered today for hisexperimental work on gravity, in particular his study ofthe equivalence of gravitational and inertial mass (theso-called weak equivalence principle) and his study ofthe gravitational gradient on the Earth’s surface. Eötvös’slaw of capillarity (weak equivalence principle) served asa basis for Einstein’s theory of relativity, and the Eötvösexperiment was cited by Albert Einstein in his 1916 pa-per “The foundation of the general theory of relativity”.(Capillarity: the property or exertion of capillary attrac-tion of repulsion, a force that is the resultant of adhesion,cohesion, and surface tension in liquids that are in con-tact with solids, causing the liquid surface to rise—orbe depressed.) The Eötvös torsion balance, an impor-tant instrument of geodesy and geophysics throughoutthe whole world, studies the Earth’s physical properties.


Figure 5. Here is the picture of the onlyauthentic relief of János Bolyai on the front of

the Culture Palace in Marosvásárhely(Romania).

famous. Unfortunately, of thisscientist-giant no picture survives,his features being forever hiddenfrom future generations. The onlysource describing his appearanceis his passport (made when he wasforty-eight): he was of averagebuild, blue-eyed and with a longface.” (Elemér Kiss)

From contemporary descriptions we may learnwhat he looked like. We know that he sported adark brown beard, that his hair was the same color,that his eyes were dark blue. According to JózsefKoncz (historian of the College of Marosvásárhely),János Bolyai looked very much like General GyörgyKlapka.7 Another important fact: his son, DénesBolyai, stated that there was a huge resemblancebetween himself and his father.

I took this thinking further. On the facade of theCulture Palace in Marosvásárhely, above the

mirror room windows, there are six carved stonereliefs of nineteenth-century intellectual geniuses.Underneath them, faded but readable subtitlesidentify each figure.

It is used for mine exploration and also in the search forminerals, such as oil, coal, and ores.7György Klapka was a heroic general of the Hungarianfreedom fight in 1848–1849.

Figure 6. The only authentic relief of JánosBolyai.

The third one from the left is Farkas Bolyai, thefourth one is János Bolyai. With the exception ofJános Bolyai, we have authentic pictures of all of theothers. I compared these pictures with the reliefs,and I found the features to be easily recognizable.

Then I looked at portraits of György Klapka andDénes Bolyai and placed them beside the JánosBolyai representation from the Culture Palace. Iwas fascinated by the likeness: as if they wereshowing the same person.

The Culture Palace in Marosvásárhely was builtbetween 1911 and 1913. At that time there werepeople living in the town who knew or saw JánosBolyai, including his son Dénes Bolyai. He was aretired judge who took part in the exhumation ofhis father and grandfather on June 7, 1911. Theartist who set János Bolyai’s features in stone atthis time must have—naturally—consulted the son(Dénes Bolyai) and his acquaintances regarding hisfather’s looks.

Help of Computer GraphicsBased on the above reasoning, we have to acceptthat there isn’t any authentic portrait of JánosBolyai. We have proved that Mór Adler’s and KárolyLühnsdorf’s pictures aren’t of János Bolyai, andthere is little likelihood of ever coming uponan authentic photo or painting in the back ofthe archives. It is very important, however, thatwe have authentic portraits of his father, FarkasBolyai, his mother, Zsuzsanna Benkö, and his son,Dénes Bolyai.

From these data, with the help of computergraphics (Meesoft SmartMorph software), Róbert


Figure 7. Computer transformation of a face: János Bolyai—Dénes Bolyai.

Figure 8. Computer transformation of a face: Dénes Bolyai—Farkas Bolyai (The likeness ofgrandson and grandfather can only be explained by the genetic mediation of János Bolyaibetween the two generations.)

Figure 9. Computer transformation of a face: György Klapka—János Bolyai.

Oláh-Gál8 and Szilárd Máté9 have created a virtualportrait of János Bolyai [20]. The aim of thisexperiment was to reduce the subjective elementin deciding which portrait is more accurate, thepicture painted by Mór Adler in 1864 or János

8Sapientia University, Department of Mathematics andInformatics, Miercurea-Ciuc, Romania.9Sapientia University, Department of Mathematics andInformatics, Miercurea-Ciuc, Romania.

Bolyai’s half-relief on the building of the CulturePalace in Marosvásárhely.

After much experimentation, using the facialtransformation technique on the computer, thefollowing conclusion was drawn: in all probability,only one of the pictures comes close to JánosBolyai’s real likeness, and it is the half-relief on theCulture Palace.

After several decades of silence, attention needsto be drawn to the fact that the face of János Bolyai


in the public consciousness is not really his face.The only authentic source of his real portrait is theCulture Palace in Marosvásárhely (Figure 6). Beinga defining figure of mathematical history, in thefuture he deserves to be associated with his realfacial features.

I present gladly to the reader the next two worksof art, which follow this mentality (see Figures 10and 11).

Furthermore, this shocking statement, referringto his appearance, may equally be applied to the“well-known” facts about his professional activity(“mind-face”).

János Bolyai’s Real “Mind-Face” as a Math-ematician (Based on E. Kiss: MathematicalGems from the Bolyai Chests [16])In his life János Bolyai’s only published work was“The absolute true science of space”, better knownas the “Appendix”. This was enough to make himworld famous, but it also reduced his intellectualcreation to this single piece of work.

When János Bolyai died the military governorseized all his manuscripts and had them put inchests and transported to the castle so that theycould be examined for military secrets. Thus werehis papers preserved for posterity, approximately14,000 pages of manuscripts. The task facingresearchers has not been easy. There are fewdates, no numbered pages, pages missing, noteson envelopes and theater programs, idiosyncraticmathematical notations, and newly invented words.

However, János Bolyai didn’t just leave us withthe “Appendix” but with a heritage, consistingof 14,000 pages of letters to his father andmanuscripts which are now kept in chests inthe Teleki-Bolyai Library in Marosvásárhely. Inthese chests one can find mathematical theories—treasures in Bolyai’s words—that have been hiddenfrom the public for nearly 100 years. These pagesconvince us that János Bolyai, who was knownpurely as a geometer, was actually a universal math-ematical genius who worked on many branchesof mathematics, at times preceding significantinventions of other big names by decades.

The task Elemér Kiss took on, deciphering thecontents of the “Bolyai chests”, led to extraordinaryresults. The expression “deciphering” describesthe tedious act of many decades by which ithas been possible to reconstruct the contents ofthese materials. The contents, the grammar, themathematical symbols, which differed significantlyfrom those of present times, were often unreadable.Today we know that the results of this hard workhave left us with a brand new “mind-face” of JánosBolyai.

Elemér Kiss’s book was published in 1999 inHungarian and in English, followed by an extendedsecond edition in 2005 by Typotex and AkadémiaPublishers [16].

Figure 10. Reconstructed portrait drawing withIndia ink, made by Attila Zsigmond (a painter

who lived in Marosvásárhely in 1927–1999),using Bolyai contemporary texts and other

sources. The picture can be found in the BolyaiMuseum, Marosvásárhely.

Figure 11. Bolyai Memorial Medal prepared forthe Bolyai anniversary (in 2002) by Kinga

Széchenyi, based on the relief on the CulturePalace in Marosvásárhely.

The first chapter, “The life of János Bolyaiand the science of space”, gives a brief accountof the journey the scientist took in the creationof a new geometry. In addition, there is a realnovelty in Chapter 1.6, considering the discoveryof non-Euclidean geometry based on facts fromBolyai’s correspondence. The author comes upwith a convincing reasoning for the priority ofBolyai in the Bolyai-Gauss-Lobacsevszky relation.

In the second chapter we can read a system-atic and comprehensive description of the “Bolyaichests”. Parts of this chapter explain the lan-guage and symbols used by Bolyai, the result ofmeticulous research, as some of the original textsresemble complicated riddles.

In Chapter 4.3 we can read that in one of Bolyai’snotes, he writes: “My long nourished expectationsand hopes had grown and mounted higher, namely,that I can devise primes based solely on their orderin their series, independently or directly …, in other


Figure 12. Elemér Kiss’s book reveals abrand-new aspect of the mathematical work ofJános Bolyai.

words, that it is possible to give a formula which willdefine only primes.”

He could find no formula for rational integerprimes and neither could anyone else till thisday, but his investigations led János Bolyai toan important discovery: he hit upon the firstpseudoprime.10

10Various composite numbers m for which the expres-sion am−1 ≡ 1 (modm) is valid. The fact that thenumber of pseudoprimes is infinite has been known since1904 [5]. There are composite numbers m which sat-isfy this congruence for each a, whenever a is a rela-tive prime to m. These numbers are called Carmichaelnumbers in honor of their discoverer [1]. One of themost recent results of number theory is the proof thatCarmichael numbers exist beyond all boundaries (basedon a 1956 idea of the magnificent Hungarian mathe-matician Paul Erdös (1913–1996)). J. Chernick proved atheorem in 1939 that can be used to construct a subsetof Carmichael numbers. The number (6k + 1)(12k +1)(18k + 1) is a Carmichael number if its three factorsare all prime. In 1994 it was shown by W. R. Alford, An-drew Granville, and Carl Pomerance that there really doexist infinitely many Carmichael numbers. Specifically,they showed that for sufficiently large n, there are at leastn2/7 Carmichael numbers between 1 and n. The researchon the pseudoprime numbers was completed in the twen-tieth century. and its more important application is incryptography [7]. I would like to add at this point thatCarmichael numbers have practical applications, namelyto attack RSA cryptosystems [21].

Bolyai believed he had discovered the formulaof primes in Fermat’s Little Theorem.11 Urged byhis father, he attempted to prove the inverse ofFermat’s Theorem, but a few attempts convincedhim that proof was impossible and the inverse ofFermat’s Little Theorem does not hold in general.12

He did not find the prime formula, but he discov-ered the first pseudoprime. He communicated thediscovery of the smallest pseudoprime (relativeto 2), 341, in a letter to his father:

“…the most immediate and propermain question, namely that it

may be the case that 2m−1

2 ≡ 1(modm), even though m is nota prime (which can of course beproven by even one example, suchas the following which I happenedto stumble on by chance, but notwithout theoretical considerations):2340 ≡ 1 (mod 341) is divisible by341 = 11 · 31, which is infinitelyeasy to ascertain from 210 = 1024,which gives a remainder of 1 whendivided by 341, therefore the re-

mainder of 21017 = 2170 = 2341−1

2

and 2341−1 ≡ 1 (mod 341) alike,thus neither the Fermat theoremnor the nice conjecture with regard

to 2m−1

2 is valid (neither in thegeneral case, nor in the particularone when a = 2), which is onlyto be regretted, since they couldhave supplied an excellent andvery comfortable new means ofidentification (criterion) of primes…”

From this letter, especially interesting is thefragment: “but not without theoretical considera-tions". What could those earlier notes include?János Bolyai examined the question under whatconditions the congruence

(1) apq−1 ≡ 1 (modpq)

is satisfied, where p and q are primes, and a isan integer not divisible by either p or q. His rea-soning was as follows, according to Fermat’s LittleTheorem: ap−1 ≡ 1 (modp) and aq−1 ≡ 1 (mod q).By raising both sides of the first congruence to the

11This theorem states that if p is a prime and a is an in-teger not divisible by p, then the difference ap−1 − 1 isdivisible by p; a usual shorthand for this is: ap−1 ≡ 1(modp).12The inverse of Fermat’s Little Theorem is: If ap−1 ≡ 1(modp) holds, it does not necessarily follow that p is aprime.


power of q − 1 and those of the second one to thepower of p− 1, we obtain:

(2)

a(p−1)(q−1) ≡ 1 (modp) and

a(p−1)(q−1) ≡ 1 (mod q)

⇒ a(p−1)(q−1) ≡ 1 (modpq).

Next Bolyai observes that if the congruenceap+q−2 ≡ 1 (modpq) = ap−1 · aq−1 ≡ (modpq)were true, then by multiplying the two expressionsobtained earlier, one could arrive at the desiredcongruence (1).

The following step must be finding the con-ditions that ensure the validity of the lattercongruence. Since ap−1 ≡ 1 (modp) and aq−1 ≡ 1(modq), continues Bolyai, there must exist integersh and k such that ap−1 = 1+hp and aq−1 = 1+ kq.In other words the condition of the validity of (1)is that

(3) hp+kq = (ap−1−1)+(aq−1−1) ≡ 0 (modpq).

It is satisfied if p is a divisor of k and q is a divisorof h, according to Bolyai, this means that apq−1 ≡ 1

(modpq) is true of primes p and q for which ap−1−1pq

and aq+1−1pq

are integers, in which case

(4)ap−1 − 1

qand

aq−1 − 1

pare also integers.

In the simple case when a = 2 Bolyai substitutessome primes satisfying (4) and arrives at p = 11and q = 31. This is how János Bolyai discoveredthe smallest pseudoprime.

Although he emphasized in his letter quotedabove that “even one example” suffices, variouscounterexamples emerge from the remainingmanuscripts. He constructed more congruences:

2340 ≡ 1 (mod 341), 414 ≡ 1 (mod 15),

2232 ≡ 1 (mod 232 + 1).(5)

Bolyai says that if in the congruence (1) is a = 2,then the congruence

(6) 2pq−1 ≡ 1 (modpq) follows.

This corresponds exactly to the theorem of JamesHopwood Jeans (1877–1940), which he publishedin 1898 [15], decades after the death of JánosBolyai. This is the case with the Jeans theoremas well. Bolyai’s discovery, like many others apartfrom the “Appendix”, was not communicated evento his father. This is why one of Bolyai’s beautifultheorems does not bear the name of János Bolyaibut that of its rediscoverer.

Bolyai aimed to extend his method (6) to thecase in which n is a multiple of three primenumbers: “…but it will be considerably more dif-ficult with three factors.” Such congruences wereconstructed by R. D. Carmichael in [1], [2]. Bolyai’sattempt suggests the following idea of generalizingJeans’s theorem: Let p1, p2, . . . , pn be primes n ≥ 1

and let a be an integer not divisible by either ofthese primes.

(7) If

ap1p2···pn−1−1 ≡ 1 (modpn)

ap1p2···pn−2pn−1 ≡ 1 (modpn−1)

···

ap2p3···pn−1pn−1 ≡ 1 (modp1)

then ap1p2···pn−1 ≡ 1 (modp1p2 · · ·pn).

Chapter 4.6 reveals that János Bolyai was alsocaptivated by Fermat numbers.13 In one of theletters written to his father, he alludes to hisapproach to Fermat numbers in two places: “Bythe way, my previous demonstration of numerusperfectus14 and that of concerning 22m +1 are goodand nice ….

“I intended to show that any number of the form2p − 1 is a prime number if p is prime, at the sametime when I took pains over 22m + 1, since as mywritings show, I thought that 2p − 1 was always aprime for any prime p. …”

This chapter represents a special value whereBolyai’s theorem on Fermat numbers is introduced.According to this “Fermat numbers are always ofthe form of 6k−1, and therefore are never divisibleby 3.” He proves the proposition as follows:

22m−1 + 1 = (2+ 1)(. . . ); consequently, 22m−1 +1 = 3n, and so 22m−1 = 3n− 1, thus 22m = 6n− 2,that is, 2 raised to an even power, is of the form6n−2. Then 22m+1 and thus 22m +1 is of the form6n− 1, with m, n ≥ 0 being natural numbers.

The international significance of this theoremand the weight of Elemér Kiss’s research have beensupported by the publication of [18], in whichTheorem 3.12 was called the “Bolyai Theorem”.

This is the first highly reputable source in whichJános Bolyai’s name is mentioned in the field ofnumber theory as opposed to geometry, which isa real milestone on the way to revealing the real“mind-face” of János Bolyai.

In Chapter 4.7, Kiss pointed out that JánosBolyai was able to prove the converse of Wilson’sTheorem,15 but he was unaware of the earlier proof

13Numbers of the form Fn = 22n + 1 where n is a natu-ral number. Fermat firmly believed that all such numberswere primes, even though he had only calculated F0 = 3,F1 = 5, F2 = 17, F3 = 257, F4 = 65637. His conjec-ture was disproven when Euler in 1732 showed that thenext Fermat number F5 = 641x6700417 is not a prime.By the early 1980s Fn was known to be composite for all5 ≤ n ≤ 32.14Perfect number.15In 1770 Edward Waring (1736–1798) announced thefollowing theorem by his former student John Wilson(1741–1793): if p is prime, then (p − 1)! ≡ −1 (modp),that is, (p − 1)! + 1 is divisible by p. The theorem


of Lagrange. Gauss discusses Wilson’s theorem inDisquisitiones Arithmeticae, but on its converse hekeeps silent. János Bolyai acquired the bulk oftheir number theoretic knowledge from the workof Gauss; thus Bolyai was unaware of the proof ofthe inverse to Wilson’s theorem.

The inverse to the theorem was important forJános Bolyai, who was interested in the prime orcomposite nature of various large numbers andwho also searched for the formula of primes. Henotes that “I have proven the inverse of the verybeautiful and significant Theorem of Wilson.” Forthe reader’s delight I present Bolyai’s proof:

Suppose that

(8) (p − 1)! ≡ −1 modp.

Let q be a prime divisor of p, namely p = q · p1,then

(9) (p− 1)! ≡ −1 mod q,

and according to Wilson’s theorem

(10) (q − 1)! ≡ −1 mod q.

It follows from (9) and (10) that(11)

(q − 1)! ≡ (p− 1)! mod q ⇒ 1 ≡ (p− 1)!

(q − 1)!mod q.

Assume now q < p, then q is a divisor of (p − 1)!but not of (q − 1)!. Hence

(12) q < p ⇒ (p− 1)!(q − 1)!

≡ 0 mod q.

The congruences (11) and (12) are contradictory;consequently the assumption q < p is false. Thusp = q is a prime.

From Chapter 4.8 we may find out that JánosBolyai obtained results on the general constructionof magic squares16 of small orders.

Bolyai wrote on the sheet containing the magicsquare that a = 3b (see Figure 13); consequentlyb = 5 is true.

At the end of his note Bolyai invites the readerto generalize his 3x3 magic square to nxn squares:

was published by E. Waring, but he acknowledged that ithad first been formulated by J. Wilson without a proof.The theorem was first proved by Joseph Louis Lagrange(1736–1813) in 1771, and he proved the inverse to Wil-son’s theorem as well: if n is a divisor of (n− 1)!+ 1, thenn is prime.16A magic square of order n is an arrangement of n2

numbers, usually distinct integers, in a square, such thatthe n numbers in all rows, all columns, and both diago-nals sum to the same constant. A normal magic squarecontains the integers from 1 to n2. The constant sum inevery row, column, and diagonal is called the magic con-stant or magic sum, a. The magic constant of a normalmagic square depends only on n and has the value a =n(n2+1)

2 .

Figure 13. János Bolyai’s general constructionof a 3x3 magic square.

Figure 14. Concrete solution of János Bolyai’s3x3 magic square.

“Seek and search the way to construct general mag-ical squares from a �17 divided into any number ofequal � − s, be it arithmetical, geometrical or har-monic progression…” I would like to add thatBolyai’s ideas were reinvented by Cayley [3] andChernick [4], and a survey on the general con-struction of magic squares can be found in anencyclopedic book [6].

Chapter 5 talks about a few results of Bolyai’swork that have never been seen before: about com-plex integers,18 into the research of which Bolyaiinvested huge amounts of energy. These studies,which Bolyai called the “theory of primes” or “imag-inary number theory”, dealt with the arithmetic ofcomplex integers.

The theory of divisibility of complex integerswas founded and developed by Gauss [12], [13].He proved the theorem corresponding to thefundamental theorem of number theory insideGaussian integers and discussed congruences in-volving complex numbers. János Bolyai elaboratedthe arithmetic of complex integers independentlyof Gauss and approximately at the same time.

17This is the original symbol on Bolyai’s sheet (see Figure13).18The complex or Gaussian integer is a complex numberwhose real and imaginary parts are both integers. For-mally, Gaussian integers are the set Z[i] = {a+bi | a,b ∈Z}, i =

√−1.


The beginnings of Bolyai’s investigation of com-plex numbers can be dated exactly, because in aletter to his father he pinpoints the date: “I soughtthe theory of imaginary quantities in their properplace and fortunately found it in 1831.” Based onhis statements it can be asserted that János Bolyaisaw his own theory clearly at the beginning of the1830s, even in the years prior to the publication ofthe “Appendix”.

In his manuscripts it can be deciphered that heclearly identified primes in the ring of complexintegers. He asserted that complex primes areeither

(13) the numbers 1+ i, 1− i, −1+ i, −1− i,(14) rational primes of the form 4m+ 3,(15) complex factors of rational primes of the

form 4m+ 1.

Bolyai merely enumerates the numbers (13) as“perfect primes”, and he notes that 2 = (1+i)(1−i),but he clearly indicates that 1+ i cannot be writtenas the product of two complex integers.

About numbers (14) Bolyai presents variousproofs that they are “absolute primes”. One of hisproofs: “If a prime p is of the form 4m + 3, thenp = t2 + u2 is impossible, because if both t and uare even or odd at the same time, then the sum oftheir squares would yield an even number and sucha number is not a prime. If one of t and u is evenand the other is odd, then the sum of their squares isa number of the form 4m+1. Then p is an absoluteprime.”

Bolyai showed that the complex integer m +ni has no divisor different from its associatesprovided thatp =m2+n2 is a prime. He relates thisproperty of numbers (15) to Fermat’s Christmastheorem, and he writes: “Every prime p of the form4m+1 is the product of two imaginary primes, sinceall such numbers are the sum of two full squares.”For example: 13 = (2+ 3i)(2− 3i).

János Bolyai also discussed unique factorizationof complex integers, and he proved the followingtheorem: “Every number of the form a + bi can beuniquely (up to the order of the factors) decomposedinto a product of finitely many primes.”

He not only elaborated the theory of complexnumbers, but its applications were also of impor-tance to him. He skillfully applied his conclusionson complex integers in proofs of various numbertheoretic theorems.

Chapter 6, “The theory of algebraic equations”,reveals Bolyai’s struggles in connection with thesolvability of algebraic equations of fifth and higherorder. His unpublished collection contains manynotes pertaining to this subject. It has beensummed up by Elemér Kiss at the end of thischapter: “János Bolyai thought long about this im-portant problem without knowing that it had beenresolved before.”

This is why the connection between Bolyai andthe theory of algebraic equations is especially in-teresting. János Bolyai frequently mentions thetwo-volume Vorlesungen über höhere Mathematikby Andreas von Ettingshausen (1796–1878), whichwas published in Vienna in 1827 (see [11]). Inthis book the author devoted an entire chapterto the impossibility of solving equations of a de-gree higher than four and cited Paolo Ruffini’s(1765–1822) proof of 179919 [22], [23]. Bolyai citesJoseph Louis Lagrange’s (1736–1813) book [19],which addresses the fundamental problem of whythe methods used for solving equations of a de-gree equal to or lower than four are inapplicablein equations of higher degree.20 On his readingBolyai writes: “…to give a proof of this impossi-bility for degree 5 and for higher degrees as well:this proof being given by Ruffini (as the deserved Et-tingshausen writes) wittily enough, but with a greatmany mistakes, in short, only in his fancies.”

This led him to conclude that the theorem wasnot valid and consequently at first, he searchedfor the solution of equations of degree higher thanfour with great enthusiasm, and he wrote in 1844:“By refuting the demonstration of impossibility (byRuffini) …it will be proven eo ipso (self-evidently) ina new way .”

János Bolyai thought long about this importantproblem without knowing that it had been resolvedbefore. On the other hand, the world didn’t knowabout this nineteenth-century Hungarian scientistwho (perhaps late and only for its own sake) hadput an end to a centuries-long debate.

The above facts suggest the isolation Bolyaiworked in all his life and the enormous creativepower through which he was able to “create a new,different world out of nothing”, not exclusively inthe field of geometry.

AcknowledgmentIt is hard to overestimate the value of the workof Elemér Kiss in the process of revealing JánosBolyai’s real face after 150 years of silence. Whatmakes it even more meaningful is the fact thatJános Bolyai’s face has been unknown so far,as the famous image that has been circulatedaround the world is certainly not his. To putan end to this misconception, and also to pub-licize essays and research in related areas, “TheReal Face of János Bolyai” has been created at:http://www.titoktan.hu/Bolyai_a.htm.

19The 1826 article by Niels Henrik Abel (1802–1829)could not have been included in this work published in1827. The Abel-Ruffini theorem states that there is nogeneral solution in radicals to polynomial equations ofdegree five or higher .20This observation not only impelled Ruffini and Abel tocontinue research in this direction but also led to Galois’sconception of group theory.


http://www.titoktan.hu/Bolyai_a.htm

George Bruce Halsted (1853–1922), Americanmathematician, translated the “Appendix” byJános Bolyai into English in 1896.

At this point it is necessary to remember theAmerican mathematician George Bruce Halsted(1853–1922), who in 1896—before any other for-eign Bolyai researcher—visited Marosvásárhely andtranslated the main work of János Bolyai: the“Appendix” [14]. With his activity he contributedsignificantly to the international appreciation ofthe two Bolyais.

I would like to sincerely thank the refereeswho supported the publication of this essay, forcontinuing the tradition initiated by G. B. Halsted.By doing so they make a significant contribution tothe introduction of János Bolyai’s real face to theworld.

This essay could not have been written withoutour personal discussions with Professor ElemérKiss, the written accounts of Professor AndrásPrékopa, and Róbert Oláh-Gál’s support with theface animation.

I would like to say a special thank you toIldikó Rákóczi, the director of the János BolyaiMathematical Society, for making it possible forme to photograph and publish pictures of keyimportance for the purposes of this project.

I would also like to express my gratitude to mydaughter Eszter and Damien Bove for taking careof the language aspect of this essay.

References[1] R. D. Carmichael, Note on a new number theory

function, Amer. Math. Soc. Bull. 16 (1910), 232–238.

[2] , On composite numbers P which satisfy theFermat congruence ap−1 ≡ 1(mod p), Amer. Math.Monthly 19 (1912), 22–27.

[3] A. Cayley, The Collected Mathematical Papers ofArthur Cayley (1889), vol. X. p. 38.

[4] J. Chernick, Solution of the general magic square,Amer. Math. Monthly 4 (1938), 172–175.

[5] M. Cipolla, Sui numeri compositi P, che verificanola congruenza di Fermat ap−1 ≡ 1(mod p), Annalidi Matematica 9 (1904), 139–160.

[6] J. Dénes and A. D. Keedwell, Latin Squaresand their Applications, Academic Press, New York,Akadémiai Kiadó, Bp., English Universities Press,London, 1974.

[7] T. Dénes, The great career of the “small Fermat the-orem" in encryption of information, Híradástech-nika, Budapest, 2002/2. pp. 59–62.

[8] , Complementary prime-sieve, Pure Math-ematics and Applications 12 (2002), no. 2, pp.197–207.

[9] , Bolyai’s treasure-chest (about the book ofElemér Kiss), Magyar Tudomány, 2006/5, 634–636.

[10] P. Erdös, On the converse of Fermat’s theorem,Amer. Math. Monthly 56 (1949), 623–624.

[11] Andreas von Ettingshausen, Vorlesungen überdie höhere Mathematik, Wiesbaden: LTR-Verlag,Neudr., 1827.

[12] C. F. Gauss, Theoria residuorum biquadratico-rum, Commentatio secunda, Göttingische gelehrteAnzeigen, Göttingen 1831, Stück 64, 625–638.

[13] , Theoria residuorum biquadraticorum, Com-mentatio secunda, Commentationes Societatis Re-giae Scientiarum Göttingensis Recentiores, Vol. VII.(1832), Göttingen, cl. math. 89–148.

[14] G. B. Halsted, The Science Absolute of Space (trans-lated from the original Latin), The Neomon, Austin,Texas, USA, 1891.

[15] J. H. Jeans, The converse of Fermat’s theorem,Messenger of Mathematics 27 (1897–1898), 174.

[16] E. Kiss, Mathematical Gems from the BolyaiChests, Akadémiai Kiadó, Budapest, Typotex LTD.,Budapest, 1999.

[17] , On a congruence by János Bolyai con-nected with pseudoprimes, Mathematica Pannonia,2004/2.

[18] Krizek-Luca-Somer, 17 Lectures on Fermat Num-bers, Springer, 2004.

[19] J. L. Lagrange, Réflexions sur la RésolutionAlgébrique des Équations, Paris, 1770.

[20] R. Oláh-Gá and Sz. Máté, Virtual portrait of JánosBolyai, 6th International Conference on AppliedInformatics, Eger, Hungary, January 27–31, 2004.

[21] R. G. E. Pinch, On using Carmichael numbers forpublic key encryption systems, Proceedings 6th IMAConference on Coding and Cryptography, Cirences-ter 1997, (ed. M. Darnell), Springer Lecture Notes inComputer Science 1355 (1997) 265–269.

[22] Paolo Ruffini, Teoria generale delle equazioni, incui si dimostra impossibile la soluzione algebraicadelle equazioni generali di grado superiore al 4◦, 2vols., Bologna, 1798.

[23] , Della soluzione delle equazioni alg. deter-minate particolari di grado sup. al 4◦, in: Mem. Soc.Ital., IX, 1802.


real face of jnos bolyai - american mathematical society

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