What does a mathematician do and why?
The conference started with why, for ten minutes. I do mathematics because I like it. We discussed briefly the distinction between pure mathematics and applied mathematics, which actually intermingle in such a way that it is impossible to define the boundary between one and the other precisely; and also the aesthetic side of mathematics. Then we did mathematics together. I started by defining prime numbers, and I recalled Euclid’s proof that there are infinitely many. Then I defined twin primes, (3,5), (5, 7), (11,13), (17,19), etc. which differ by 2. Is there an infinite number of those? No one knows, even though one conjectures that the answer is yes. I gave heuristic arguments describing the expected density of such primes. Why don’t you try to prove it? The question is one of the big unsolved problems of mathematics.
KeywordsPrime Number Infinite Number Pure Mathematic Asymptotic Relation Riemann Hypothesis
Unable to display preview. Download preview PDF.
- V. Brun, “Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare,” Archiv for Mathematik (Christiania) 34 Part 2 (1915), pp. 1–15.Google Scholar
- G.H. Hardy, A Mathematician’s Apology, Cambridge University Press, 1969.Google Scholar
- G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Oxford University Press, 1980.Google Scholar
- A.E. Ingham, The Distribution of Prime Numbers, Hafner Publishing Company, New York, 1971 (Reprinted from Cambridge University Press).Google Scholar
- S. Lang and H. Trotter, Frobenius Distributions in GL 2-extensions, Springer Lecture Notes 504, Springer-Verlag, New York, 1976.Google Scholar
- J.J. Sylvester, “On the partition of an even number into two prime numbers,” Nature, 55 (1896-1897), pp. 196–197 (= Math. Papers 4, pp. 734-737).Google Scholar
- D. Zagier, “The first 50 million prime numbers,” Mathematical Intelligencer, 1978.Google Scholar