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Old and new unsolved problems in plane geometry and number theory. (English) Zbl 0784.51002
The Dolciani Mathematical Expositions. 11. Washington, DC: Mathematical Association of America. xv, 333 p. (1991).

24 central problems are presented, each accompanied by a collection of other related problems and a survey on partial solutions. For each problem, the exhibition is divided up into three sections. The first one gives an elementary overview, discussing the history as well as solved and unsolved variants of the problem. The second one contains more details, a wider and deeper survey of what is known about the problem, and rather comprehensive collection of references. The third part presents the solutions to the excercises which have been formulated in the preceding two parts.

The first 12 problems deal with two-dimensional (and higher-dimensional) geometry. Problem 1: Is each reflecting polygonal region illuminable? Here also periodic as well as dense billiard paths on polygonal billiard tables are discussed. See e.g. F. B. Holt [Geom. Dedicata 46, 73-90 (1993; Zbl 0784.51018)] for some recent achievements in this direction. Problem 2 deals with the existence of two equichordal points for a simply closed curve. The presentation also includes a hint to recent work of R. Schäfke and H. Volkmer [Asymptotic analysis of the equichordal problem, J. Reine Angew. Math., to appear] where in the case of plane curves the nonexistence of these two points has been established up to a finite number of eccentricities. Problem 3: When congruent disks are pushed closer together, can the area of their union increase? Problem 4: If a convex body $C$ contains a translate of each plane set of unit diameter, how small can $C$’s area be? Problem 5: How many points are needed to guarantee a convex $n$-gon? Problem 6: If $n$ points are not collinear, must one of them lie on at least $n/3$ connecting lines?

Under Problem 7 a collection of several problems dealing with tilings of the plane is listed and discussed. Problem 8: What is the minimum of colors for painting the plane so that no two points at unit distance receive the same color? Problem 9: Can a circle be decomposed into finitely many sets that can be rearranged to form a square? In a footnote to this problem the authors mention that it has been solved affirmatively by M. Laczkovich [J. Reine Angew. Math. 404, 77-117 (1990; Zbl 0748.51017)] while this book was going to press. Problem 10: Does the plane contain a dense rational set? According to the authors’ definition, in a rational set only rational distances can be attained. Problem 11: Does every simple closed curve in the plane contain all four vertices of some square? Here also other inscribed polygons are discussed. For the case of inscribed triangles recent progress was obtained in a paper by M. J. Nielsen, Geom. Dedicata 43, No. 3, 291-297 (1992; Zbl 0767.51009).

Problem 12 deals with a famous topic from topology: Does each nonseparating plane continuum have the fixed-point property?

The subjects of the next 8 problems belong to number theory. Problem 13 deals with Fermat’s Last Theorem. The proof of this theorem is one of the notorious problems in number theory. Consequently the main subject of this section is the presentation of a comprehensive survey on approaches to a solution of this problem. Of the same type are Problem 17 (the Riemann hypothesis) and Problem 20 (Hilbert’s Tenth Problem): Is there an algorithm to decide wether a polynomial with integer coefficients has a rotational root? In Problems 13 and 20 as well as in some of the subsequent problems special attention is given to algorithmic questions.

Problem 14: Does there exist a box with integer sides such that the three face diagonals and the main diagonal all have integer lengths? Problem 15: Does the greedy algorithm always succeed in expressing a fraction with odd denominator as a sum of unit fraction with odd denominator? Problem 16 deals with perfect numbers, Mersenne primes and twin primes. In Problem 18 algorithms for primality testing and the computation of the prime factorization are discussed. Problem 19 studies the behaviour of the iterations of the function which sends each natural number $n$ to $\frac{n}{2}$ in the even case and to $3n+1$ in the odd case.

The last four problems of this collection deal with interesting real numbers. These include the questions of patterns in the decimal expansion of $\pi$, the algebraic dependence of $\pi$ and $e$, the real-time computability of irrational numbers, and the irrationality of the limits of certain types of series.

The book will be found fascinating and useful by a wide range of mathematicians – teachers, undergraduate and graduate students as well as researchers. It will provide a good background for courses in geometry or number theory. Clearly, it will be extremely useful as a text for a seminar about unsolved problems.

##### MSC:
 51-02 Research monographs (geometry) 11-02 Research monographs (number theory) 52-02 Research monographs (convex and discrete geometry) 52C20 Tilings in 2 dimensions (discrete geometry) 51M20 Polyhedra and polytopes; regular figures, division of spaces 00A07 Problem books 51M16 Inequalities and extremum problems (geometry) 52A40 Inequalities and extremum problems (convex geometry)
##### Keywords:
plane geometry; number theory; problems