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Georg Friedrich Bernhard Riemann (1826--1866) --- Historical Sketch

Georg Friedrich Bernhard Riemann's name appears in every calculus course because of the Riemann sum. But the man for whom it is named did far more than that. A conjecture he made in 1859 is today the most famous unsolved problem in mathematics, the Riemann hypothesis. (There is a Clay Mathematics Institute$^{\text{*}}$ prize of a million dollars for the person who first solves it.)

Riemann was born in Germany in September 1826, his father Friedrich a Lutheran minister. He was second born in a family of six children. The father tutored the children, and when the young Bernhard was ten, a local teacher helped. Starting at age 14, he continued in public education and took a particular liking to mathematics. He read a book of some 900 pages on number theory in but six days.

At age 19, Bernhard entered the University of Göttingen, moving two years later to the University of Berlin. Then in 1849, he moved back to Göttingen. earning his doctorate under Carl Friedrich Gauss in 1851. Eight years later, he was appointed to the chair of mathematics at Göttingen, and within a few days was elected to the Berlin Academy of Sciences, where he sent a report entitled On the number of primes less than a given magnitude. In that paper appeared Riemann's zeta functionMATHwhich can also be written as the infinite productMATHwhere $p_{i}$ is the $i$th prime. Dealing with this second representation, Riemann made a conjecture as to where all of the roots of MATH lie in the complex plane, and it is this problem that was mentioned in the first paragraph of this sketch.

Here are some problems having to do with prime numbers, which seems fitting in this sketch.

1.

A prime number is called a Mersenne prime if it can be written as $2^{n}-1$ for $n>1$ an integer. Name the first four Mersenne primes.

2.

In the first four Mersenne primes, the exponent on $2$ is itself a prime. Is this an accident?

3.

A prime number is called a Fermat prime if it can be written as MATH for $n\geq 0$. Pierre de Fermat conjectured that all numbers of that form are prime, but Euler proved that MATHWhat are the first four Fermat primes?

$^{\text{*}}$The Clay Mathematics Institute was founded in 1998 by Mr. Landon T. Clay and his wife Lavinia D. Clay. The aim of the Institute is "to increase and disseminate mathematical knowledge."


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Historical sketches

Riemann sum


Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.

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