Mathematics in 10 Lessons: The Grand Tour by King Jerry P

Author:King, Jerry P. [King, Jerry P.]
Language: eng
Format: epub
Publisher: Prometheus Books - A
Published: 2009-04-01T16:00:00+00:00

CONJECTURES

Classical number theory abounds with conjectures and open problems. Many of these are ancient enough and deep enough to bring to their conqueror mathematical fame.

Here are three open problems:

(1) Twin prime : If p is a prime number then p + 1 can be prime only if p = 2. (Each prime p other than 2 must be odd. So, p + 1 is even and cannot be prime.) But p and p + 2 might be prime. Examples are 3 and 5, 5 and 7, 11 and 13, 17 and 19. Such pairs of primes are called twin primes. How many twin primes are there? No one knows. Probably there are infinitely many but no proof of this exists.

(2) Perfect numbers : A natural number n is perfect if n equals the sum of its proper divisors. Examples are 6 and 28:

6 = 1 + 2 + 3

and

28 = 1 + 2 + 4 + 7 + 14.

The next two perfect numbers are 496 and 8,128.

How many perfect numbers are there? Do odd perfect numbers exist? No one knows.

(3) The Goldbach conjecture : In 1749, Christian Goldbach produced the conjecture:

Every even integer greater than 4 can be written as the sum of two odd primes.

For example,

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