**Elemér Kiss**was born in Brașov, Romania. The town is also known as Kronstadt in German and Brassó in Hungarian. His father was István Kiss (1904-1996) and his mother was Berta Csedő (1904-1985). He was brought up in Armaseni (Hungarian name Csikménaság), a village in Csík region, where his father was a teacher in the local school and his mother was a kindergarten teacher. Their family, as many families at that time, led a very simple life, but they brought up their three children with great love, all of them graduating from university. His father, István, was a beekeeper as well as a school teacher, and the respect for nature and local region that Elemér inherited from his parents was evident later on in papers he wrote on bees, namely

*The Bees and Geometry*and

*Construction of Bees*. He attended high school in Miercurea Ciuc (Csíkszereda) where he studied at the Gymnasium where his mathematics teachers were Gergely Borcsa and Kálmán Bakó. He began his studies at the Catholic Gymnasium in 1940 and, after eight years at the school, Kiss went to Cluj where he attended the Bolyai University, beginning his studies in 1948. There is a quite complicated story of the Hungarian and Romanian universities in Cluj (and even the name of the city of Cluj, which was renamed Cluj-Napoca in the Ceausescu era). For a short history of the Bolyai University we refer the reader to our biography of Gyula Maurer who was also a student at the Bolyai University when Kiss entered, but two years ahead of him. At the Bolyai University Kiss had Samu Borbély, Ferenc Radó and Jenő Gergely among his professors.

Kiss graduated from the Bolyai University in 1951, obtaining a qualification to teach mathematics and physics at a high school, and spent the next ten years as a secondary school teacher. He was employed at the Farkas Bolyai High School in Târgu-Mureș (the city is known as Marosvásárhely in Hungarian). This school is named after the mathematician Farkas Bolyai who was a teacher at the school. His son János Bolyai, an even more famous mathematician than his father, also studied at this school. However, this school has a long history stretching back to 1557 and was one of the first schools of the Reformed Church in Transylvania. Kiss taught at this famous school until 1961 when he was appointed as a lecturer in the Mathematics Department of the Teachers' College in Târgu-Mureș. This Pedagogical Institute had been founded in Târgu-Mureș in 1960 and, despite several changes of name and changes in structure, has continued to operate in the city being, since 1996, the Petru Maior University of Târgu-Mureș. We note that Petru Maior (1760-1821) after whom the university is now named, was one of the leading Transylvanian scholars of his time making contributions to history, philosophy, linguistics, Christian morals, and to both secular and religious education. In this period, Kiss was elected a member in the leadership of the Romanian Society of mathematics between 1962-65.

In 1964 he was married to Ágnes Julianna Tankó, a mathematics teacher, and they had two sons, Levente, born in 1966, and Péter born in 1969. Levente became a scientist in Budapest, he was awarded a D.Sc. in biology, and has four children, Márton, Boróka, Kornél-Ábel, and Lotti-Emma, while Péter was awarded a Ph.D. in electrical engineering. He now lives in New Jersey and has three daughters, Zsuzsa, Eszter and Kinga.

In 1974 Kiss was awarded his doctorate from the Babeș-Bolyai University of Cluj for his thesis on algebra. His thesis advisor was the algebraist Gheorghe Pic who was head of the Department of Mathematics at Babeș-Bolyai University. Kiss served as Head of the Department of Mathematics between 1976 and 1985. At this time the institution in which he worked was called the Institute of Higher Education of Târgu-Mureș. In 1985 it became the Institute of Education and Engineering, being part of the Polytechnic Institute of Cluj, then in 1991 it became the Technical University of Târgu-Mureș. Kiss continued to teach at this institution in each of its different characters and different names. Between 1975 and 1995 he was a member of the editorial board of the mathematical journal *Matematikai Lapok*, edited in Cluj. In 1999, when Kiss was seventy years of age, he retired from the Petru Maior University of Târgu-Mureș. At this time he was one of the founders of the Sapientia Hungarian University of Transylvania in Târgu-Mureș where he worked until his death.

We look now at the mathematical contributions made by Kiss. His first papers were on algebraic equations and polynomials. He published two papers in Romanian, namely *Common roots of an arbitrary number of algebraic equations of higher order* (1966) and *A theorem of Hurwitz type* (1967). The first of these gives theorems specifying necessary and sufficient conditions for *n* - *k* equations each of degree *n* to have *k* + 1 common roots, 0 ≤ *k* ≤ *n* - 1. The second paper generalises a theorem of Hurwitz giving conditions on the coefficients of a polynomial under which the zeros lie within a certain annulus in the complex plane. The next topic to interest Kiss was related to ring theory. He published two papers in German, namely *Über eine Kategorie von Halbmoduln* Ⓣ (1974) and *Über das Linkspseudozentrum eines Ringes* Ⓣ (1978). In the second of these papers Kiss generalises standard properties of the centre of a ring to the pseudocentre. He continued his work on the pseudocentre of a ring and published two papers in English, namely *Rings with dense left pseudocenter* (1988), and *Left pseudocommutative rings* (1988).

The research for which Kiss is most famous is his contributions to the history of mathematics which he began in the middle of the 1990s. His contributions to the history of mathematics are exclusively on topic related to János Bolyai. In particular he studied number theory and algebra in Bolyai's work leading to a much better understanding of the range of Bolyai's contributions. Before this work, almost everyone had thought that Bolyai's only significant contribution was to non-euclidean geometry. In 1995 Kiss published *Fermat's theorem in János Bolyai's manuscripts*. Working on Bolyai's manuscripts, Kiss found that Bolyai investigated Fermat's Little Theorem which, he says, is "unexpected, for all the authors of Bolyai's monographs agree with the opinion that the creator of the non-Euclidean geometry hadn't ever been dealing with number theoretic problems".

Kiss continued to explore this line of research and published *A short proof of Fermat's two-square theorem given by János Bolyai* (1997), *János Bolyai's enquiries on the decomposition of the primes *4*m* + 1* into sums of two squares* (1998), and *Notes on János Bolyai's researches in number theory* (1999). Stefan Porubsky writes in a review of the 1999 paper:-

In 1999 Kiss's bookJános Bolyai is mostly known as one of the founders of non-Euclidean geometry. It is less well known that he left several thousand pages of manuscript notes written for his own satisfaction. After examining the manuscripts kept in the Teleki-Bolyai Library in Marosvásárhely(Târgu-Mureș), the author reviews the number-theoretical achievements of Bolyai in this paper. It is documented here that Bolyai's manuscripts contain many ideas which have hitherto been attributed to other mathematicians. The topics covered are related to the converse of Fermat's little theorem, pseudoprimes to base2, complex integers, decomposition of primes as the sum of two squares and Fermat numbers.

*Mathematical gems from the Bolyai chests. János Bolyai's discoveries in number theory and algebra as recently deciphered from his manuscripts*was published. This was an English translation of his Hungarian text. The book received an award from the Hungarian Academy of Science for its exceptional value and, in addition, received the Bolyai Award of the János Bolyai Foundation. Colin Fletcher reviewed this important book and, after giving a summary of Bolyai's life, writes:-

After the publication of this book, Kiss continued to publish papers improving understanding of the mathematical achievements of János Bolyai. He publishedSome biographers have suggested that his[Bolyai's]originality was dissipated after the appearance of the Appendix, and even that his interest in mathematics slumped. Using the fields of algebra and number theory this book attempts to show that this was not so. When János Bolyai died, the military governor seized all his manuscripts, and had them put in chests and transported to the castle so that they could be examined for military secrets. Thus were his papers(by and large)preserved for posterity, approximately14,000pages of manuscripts. The task facing researchers has not been easy: there are few dates, no numbered pages, pages missing, notes on envelopes and theatre programmes, idiosyncratic mathematical notations and newly invented words. Examples of the mess are given in the text. It is known that a number of letters and notes have been destroyed.1837

Bolyai's "Responsio" was his entry to a Leipzig competition ofconcerning the construction of complex numbers. In it he defined them as ordered pairs of reals(as had Hamilton a few years earlier)and he made a connection with the non-Euclidean geometry of the Appendix. In number theory a lot of his work was original, some of it preceding published work of others. He was interested in prime numbers both inZand inZ[i]. InZhe tried to find a formula for primes and was led to speculate whether Fermat's theorem had a converse. This in turn led him to the discovery of the smallest pseudoprime341, and to the proof of a theorem of Sir James Jeans which was published while he was an undergraduate in1898.4

Bolyai displayed an odd mixture of ignorance and originality. He was able to give new short proofs of Fermat's theorem on primes of the formm+ 1, and yet at one point he assumed that2^{p}- 1was always prime. His knowledge of number theory was mainly derived from the 'Disquisitiones' of Gauss, and this produced blind spots. Gauss did not mention that the converse of Wilson's theorem was proved by Lagrange, and so Bolyai was unaware of the fact and he produced his own proof.

Turning to Bolyai's work on the Gaussian integers, the author claims that Bolyai "elaborated the arithmetics of complex integers independently of Gauss, and approximately at the same time". Gauss still had the priority and indeed the more exhaustive treatment, but Bolyai independently had the results on greatest common divisors and unique factorisation. Earlier historians, such as Stäckel, seem to have missed such originality.

Finally, we note that Bolyai worked on the solvability of polynomial equations and the fundamental theorem of algebra. The text ends with translations of a number of letters between father and son. Our one-club golfer has turned out to be more of an all-round player.

*János Bolyai, the versatile mathematician*(2002), and (with József Sándor)

*On a congruence by János Bolyai, connected with pseudoprimes*(2004). In the second of these papers the authors study a congruence discovered in manuscripts of Bolyai. They prove some theorems related with pseudoprimes. These theorems give necessary and sufficient conditions to obtain certain congruences modulo a power of a prime.

In 2006, the year in which Kiss died, he published *János Bolyai's new face*. Doru Stefanescu writes in a review:-

Four years after Kiss's death, a joint paper with Péter Gábor Szabó was published, namelyThe paper presents the main mathematical contributions of János Bolyai in number theory and algebra. The author studied several manuscripts of Bolyai and found that Bolyai obtained consistent results in fields other than geometry. In the paper under review, the author discusses the results of Bolyai on Fermat's little theorem, pseudoprime and Mersenne numbers, and Diophantine and exponential Diophantine equations. The paper contains many excerpts from the manuscripts of Bolyai and a short section on his correspondence with his father Farkas Bolyai, who was also a mathematician.

*The problems of the mathematical analysis in the manuscript heritage of the two Bolyais*(Hungarian) (2010).

Among the honours given to Kiss we mention his election to the Hungarian Academy of Sciences (2001) and being made a Freeman of the city of Târgu-Mureș (2006). The Kiss Elemér College of the Sapientia Hungarian University of Transylvania in Târgu-Mureș is named for him.

**Article by:** *J J O'Connor* and *E F Robertson*

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