Saturday, March 14, 2009

Sierpinski numbers

A Sierpinski number is an odd positive integer k such that all integers of the form k2n + 1 (with n taking positive integer values) are composite. In [1], Sierpinski showed that there are infinitely many k with this property. The "Sierpinski problem" asks:
" What is the smallest Sierpinski number ? "
It is conjectured that the answer is k = 78557 (the fact that 78557 is a Sierpinski number was discovered in 1962 by John Selfridge).

Waclaw Sierpinski was born on March 14, 1882 in Warsaw.
[1] Sierpiński, Waclaw "Sur un problème concernant les nombres k2n + 1." Elemente der Mathematik 15, 73-74 (1960).

Friday, March 6, 2009

Ferdinand von Lindemann (1852 - 1939)

Carl Louis Ferdinand von Lindemann died 70 years ago. In 1882 he provided the first proof of the transcendence of Pi - in other words, Pi is not a root of any non-zero polynomial with rational coefficients. As a corollary, this provided a negative answer to the "squaring the circle" problem (one of the oldest geometry problems, asking whether it's possible to construct a square with the same area as a given circle by using only a finite number of steps involving compass and straightedge).

For an informal and very brief introduction to the term "transcendental number" we can start by first defining an algebraic number to be a number - real or complex - that is a root of some non-zero polynomial with rational coefficients. For example,

is algebraic, since it is (you check!) a root of the equation

One could say that there are "very few" algebraic numbers - technically they form a countable "measure-zero subset" of the real number line (or of the complex plane, if we speak about complex algebraic numbers). The transcendental numbers (whether real or complex) are precisely the numbers that are not algebraic. Here we can say just the opposite: "almost all" numbers are transcendental. Intuitively, if we imagine a point-like "shuttle" landing at random on some interval "planet" [a,b] with b > a , then the probability of the landing site being transcendental is 1. The tricky part is that even if "almost all" numbers are transcendental, it may be very hard to prove that a given number is transcendental! The study of transcendental numbers involves very sophisticated, elegant mathematics.