New Looks at Old Number Theory

AbstractWe present three results of number theory that all have classical roots, but also modern aspects. We show how to (1) systematically count the rational numbers by iterating a simple function, (2) find a representation of any prime congruent to 1 modulo 4 as a sum of two squares by using simple properties of involutions and pairs of involutions, and (3) find counterexamples to Euler's conjecture that a fourth power can never be the sum of three fourth powers by using properties of quadratic polynomials with rational coefficients. Additional informationNotes on contributorsAimeric MalterAIMERIC MALTER is a high school student in Bremerhaven, Germany. At age 13, he was the youngest participant at the International Mathematical Summer School for Students in Bremen in 2011. He enjoyed the presentations and the interesting people he met there and hopes to be invited again in the future. One of the talks by Don Zagier inspired him to carry out a research project for the German student competition “Jugend forscht,” where he won the first prize on the junior level. One year after the summer school, he completed the mathematics part of the high school curriculum. Aimeric likes to go swimming, indoors and outdoors.Dierk SchleicherDIERK SCHLEICHER is professor of mathematics at Jacobs University Bremen. Much of his research is on dynamical systems, especially the theory of iteration and complex dynamics. He enjoys the international spirit in mathematical research; before and after his Ph.D. at Cornell University, he spent longer periods of time at Princeton, Berkeley, Paris,München, Toronto, and Providence, and many shorter ones in Russia and elsewhere. One of his main professional goals is to bring together leading mathematicians of today and tomorrow, for instance by organizing events such as the 50th anniversary of the International Mathematical Olympiad, and of course summer schools such as the one that brought together his co-authors Don and Aimeric. In his free time, he enjoys outdoor activities such as kayaking, sailing, paragliding, and outdoor swimming.Don ZagierDON ZAGIER is a scientific member and director of the Max Planck Institute for Mathematics in Bonn and professor of number theory at the Collège de France in Paris. His mathematical interests center around number theory (especially the theory of modular forms) and its applications in topology, algebraic geometry, and mathematical physics, but he is happy to work on any problem that involves enough computation. Having lived in more than half a dozen countries, and having fallen in love with mathematics and gone through school and university at a very early age, he too is very enthusiastic about activitues that encourage the love of mathematics in young students and that bring together people from different countries. His main hobbies are languages and piano; his favorite sports, skiing and sudoku.