Prof. RNDr. Břetislav Novák, DrSc.

A nascent mathematician

The mathematical talent of Břetislav Novák started to manifest already during his school-days and demonstrated itself outwardly in mathematical olympiad and almost simultaneously in his first papers.

In the former Czechoslovakia the mathematics competition known as Mathematical Olympiad was organized annually since the school year 1951/52. Mathematical Olympiad (MO) was the largest organized mathematics competition in former Czechoslovakia. Its aims was the discovering, encouraging and challenging of mathematically gifted students of age from 13 to 18 years. In the first two years it was organized only for high school students in three categories A (the highest category for grade 11), B (grade 10), and C (grade 9). Thus the category C was designated for the first grade of high schools. Starting with the 3^rd MO (school-year 1953/54) also the category D for the highest class of the 2^nd stage of the compulsory elementary schools was established.

The selection process started with a school round in each category. After successfully completing the first school round the successful participants were invited to the district (regional) round. In the highest category A the best contestants of the regional rounds were in addition invited to the third national round. The contestant with at least 1/2 of scores of the round became the so called successful solver.

Břetislav started his mathematical career as an Olympian in the school-year 1953/54 in category C. He lived in historically important Chrudim, a town in Pardubice region in Eastern Bohemia. In this year there were only 6 contestants in the first round of category C in the whole Pardubice region, from which two qualified into the second round but only one - Břetislav participated on the second round and also became the only successful solver.

In the school year 1954/55, that is in the 4^th MO there were 53 contestants in the regional 1^st round of category B from which only 12 advanced the 2^nd round. In the second round there were 3 contestants but the absolute winner was Břetislav Novák.

In the 5^th MO in the school-year 1955/56 there were 32 contestants in the first round of the category A in Pardubice region from which struggled in the second round only 6 and here only 3 qualified for the 3^rd national round. Since 185 successful solvers of the regional rounds qualified into the 3^rd national round, a special committee selected 80 from them for the national round from which only came 77. The absolute winner became Břetislav.

The Central Committee of the MO issued every year a booklet containing comprehensive information about all categories including problems posed in all categories and rounds together with their solutions. If some of the contestants found an interesting solution, the solution was reproduced here. For instance, a part of the Břetislav solution of the 7^th problem of the 1^st round of category A can be found in the booklet devoted to the 5^th MO.

Simultaneously with the events described above, the young Břetislav followed also Czech mathematical journals, among others the Časopis pro pěstování matematiky (Journal for Cultivation of Mathematics) ¹  later often named by Břetislav by its generally accepted nickname Pěstouš. In 1953 appeared here a small note ²  by J.Mařík on quadratic polynomials taking prime values at many values of the argument (This note was motivated by a letter, by concurrence of circumstance, by some Mrs.Novák ³ from Prague 2 to Prof.Bydžovský⁴. ) Since every prime number >3 is of the form 6m±1, in his first short paper ^{5, 6}  Břetislav characterized quadratic polynomials with integral coefficients taking at integral values only values of the form 6m±1. The kernel of his proof is the observation that if a quadratic polynomial with integral coefficients takes at all integral values only values of the form 6m±1 at six consecutive values of the argument, then all its values are of the form 6m±1. In his second paper ⁷  he improves this argument proving that if an arbitrary polynomial with integral coefficients takes only values of the form 6m±1 at three consecutive values of the argument, then all its values are of the form 6m±1.

Recollections on his local teachers of mathematics S.Novotný and J.Hamerlík in Chrudim and the head organizer of the MO in Pardubice region J.Honzák can be found in the booklet 25 years of Mathematical Olympiad in Czechoslovakia ⁸. As it is written here, to the first named teachers is to thank that  in the Eastern Bohemian Athens ⁹ grew up a plentiful group of young people for which the love to mathematics remained a livelong illness.

B.Novák followed course of events around the MO for the rest of his life. For instance, his last unfinished manuscript was motivated by a problem posed at the 31^st International Mathematical Olympiad held in Beijing, China (July 8 - July 19, 1990). Here the problem sent by Romania required ¹⁰ :  Determine all integers n>1 such that (2ⁿ+1)/n² is an integer. It follows from the solution that the least prime divisor of a solution n is p=3 (actually the only solution is n=3.)