OLD AND NEW PROBLEMS AND RESULTS IN COMBINATORIAL NUMBER THEORY : van der WAERDEN'S THEOREM AND RELATED TOPICS

The present paper represents essentially a chapter in a forthcoming "Monographie" in the l Enseignement Mathématique series i ) with the title "Old and new problems and results in combinatorial number theory" by the above authors . Basically we will discuss various problems in elementary number theory, most of which have a combinatorial flavor . In general we will avoid classical problems, just mentioning references for the interested reader. We will almost never give proofs but on the other hand we will try to give as exact references as we can . We will restrict ourselves mostly to problems on which we worked for two reasons : (i) In order not to make the paper too long ; (ii) We may know more about them than the reader . Both the difficulty and importance of the problems discussed are very variable-some are only exercises while others are very difficult or even hopeless and may have important consequences or their eventual solution may lead to important advances and the discovery of new methods . Some of the problems we think are difficult may turn out to be trivial after all -this has certainly happened before in the history of the world with anyone who tried to predict the future . Here is an amusing case . Hilbert lectured in the early 1920's on problems in mathematics and said something like this-probably all of us will see the proof of the Riemann hypothesis, some of us (but probably not I) will see the proof of Fermat's last theorem, but none of us will see the proof that 2 ,/2 is transcendental . In the audience was Siegel, whose deep research contributed decisively to the proof by Kusmin a few years later of the transcendence of 2 ,/2 . In fact shortly thereafter Gelfond and a few weeks later Schneider independently proved that cc" is transcendental if a and /3 are algebraic, /3 is irrational and a # 0, 1 .