Introduction: The Ever Growing Nature of Mathematics

Beyond its glorious past, mathematics has a vivid present and a promising future. New results are published on a regular basis in the professional journals; new problems are created and added to a plethora of yet unsolved problems, which challenge mathematicians and occupy their minds.

Movshovitz-Hadar [13] suggested a classification of mathematical news into five categories which we bring here with examples, many of which can be made accessible to high-school students:

1. (i) A recently presented problem of particular interest and possibly its solution. E.g. Herzberg and Murty's paper concerning the mathematical problems related to Sudoku puzzles [10] and Murty's later discovery of a Sudoku puzzle with exactly two solutions [14].

2. (ii) Long-term open problems recently solved. “Recently solved” is defined as past 30 years and “long-term” is defined as at least 100 years. E.g. The proof of Kepler's conjecture; The mapping of the E8 group; The Four Color Problem; Fermat's Last Theorem. For a more comprehensive (yet partial) list see Movshovitz-Hadar [13].

3. (iii) A recently revisited problem. This category includes a new proof to a known theorem, or new findings in an already solved problem, or a new solution to a previously solved problem, or a generalization of a well established fact, or even a salvaged error. E.g., In 1996, Robertson, Sanders, Seymour and Thomas published a proof of the Four-Color problem, freeing it from the doubts about its computer-assisted proof, provided twenty years earlier by Appel and Haken [19]. Another example is the constant race for new prime numbers, elaborated in Section 9.6, below.

4. […]