Ancient Greeks: Prime Numbers and Number Theory Michael Lahanas Αρχαίοι 'Ελληνες : πρώτοι αριθμοί, θεωρία αριθμών

Griechische Mathematik: Zahlentheorie und Primzahlen

(Θυμαρίδας ο Πάριος) a Pythagorean a number theorist called a prime number rectilinear since it can only be represented one-dimensionally, whereas non-prime numbers such as 6 can be represented by rectangles of sides 2 and 3. He is known also for his epanthema (flower of Thymaridas)

For large numbers such as 7380563434803675764348389657688547618099807 the theorem tells us that it has a prime factorization. It is easy to write a little program that produces the prime factorization for this number and for any arbitrary large integer. However, the average time such a program would take to find the prime factorization of an integer n goes up dramatically as n gets large: for sufficiently large n, even the fastest super-computer available today would —on average—take longer to find the prime factorization of n than the age of the universe. So, although a prime factorization of a positive integer always exists, it may be impossibly hard to find. In fact, this is a good thing—it is at the heart of the public key codes that make credit card transactions on the Internet safe.

Euclid's Algorithm

Euclid of Alexandria (Ευκλείδης ο Αλεξανδρεύς) provided the first algorithm for the calculation of the greatest common divisor gcd(a,b) of two integers a and b. This algorithm belongs to the so called class of integer relation algorithms which for a given set of real numbers (x₁,x₂,...,x_n) try to find if integers a₁, a₂, ..., a_n (not all 0) exists such that:

a₁x₁ + a₂x₂ + ...+a_nx_n = 0.

) (from http://www.larouchepub.com/)

Perfect Numbers

Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist.
St. Augustine (354-430) in The City of God, who according to Bernal (1965) was "turning from wicked learning to holy nonsense.." Numerology in the Bible: "Bring some of the fish you've just caught," Jesus said. So Simon Peter went aboard and dragged the net to the shore. There were 153 large fish, and yet the net hadn't torn. )

The Pythagoreans were fascinated by whole numbers. They defined as perfect numbers those equal to the sum of their parts (or proper divisors, including 1). For example, 6 is the smallest perfect number--the sum of its three proper divisors: 1, 2, and 3. The next perfect number is 28, which is the sum of 1, 2, 4, 7, and 14.

If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. Euclid Elements Book IX , proposition 36

The first four perfect numbers are: 6, 28, 496, 8128. Euclid was able to find that each of these numbers is of the form 2ⁿ(2ⁿ⁺¹- 1), where 2ⁿ⁺¹-1 is prime. Euclid proved that all numbers of this form were perfect. The Pythagoreans knew that 1 + 2 + 4 + ... + 2^k = 2^k+1 - 1.

Euclid's perfect numbers are G point on the line". I have no direct information of a text by the ancient Greeks which discusses the Fibonacci numbers but I found the following information interesting (

, Joseph Henry Press (April 23, 2003)

, Narkiewicz , Wladyslaw , Springer 2000, Springer Monographs in Mathematics (the development of Prime Number Theory from its beginnings until the end of the first decade of the 20th century)

Tom M. Apostol, Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics) , Springer 1998