Euclidean Number theory

From the Algorithm to the Basic Theory of Numbers

The Life of Euclid

• Born in Alexandria around 300 B.C.

• Studied under Plato and taught in the Museum

• Little known about his life

• Well kept records with team of mathematicians

• Wrote “The Elements”

It’s not all geometry!

• The Elements made up 13 books mostly on geometry and algebra

• Logical sequence made the elements useful• Books 7-9 dealt with number theory• The opening of a new way of looking at

math• The Euclidean Algorithm is introduced in

book seven

The Euclidean Algorithm

• Used to find the g.c.d of two numbers• Method: divide the larger of the two positive

integers by the smaller one, then divide the divisor by the remainder. Continue this process, of dividing the last divisor by the last remainder, until the division is exact. The final divisor is the sought g.c.d. of the two original positive integers.

Find the g.c.d of 63 and 24.

236

3_169

6_1915

9_11524

15_22463

R

R

R

R

•So the g.c.d. of 63 and 24 is 3!    

Just the beginning of number theory

• Euclid dives into the concept of prime numbers

• Introduction of relatively prime numbers

• Modern mathematics is still trying to discover more about primes

• Theory proofs all based on previously proven information

Problems from Eves

• 5.2 a.) Prove, using Problem Study 5.1 (f)*, that if p is a prime and divides the product of uv, then either p divides u or p divides v.

• *5.1 (f) states that 2 integers a and b are relatively prime if and only if there exist integers p and q such that pa+qb=1

Assume p does not divide u:

• Then p and u are relatively prime

• So pa+ub=1• pav+ubv=v (multiply through by v)

• px=uv (since p divides uv, some x exists such that px=uv)

• pav+bpx=v

• p(av+bx)=v

Since p(av+bx)=v:

• av+bx is an integer (since integers are closed under addition and multiplication)

• So av+bx can be written as some integer h.

• ph=v

• Therefore p must divide v!

Fundamental theorem of Arithmetic

• Every integer greater than one can be uniquely factored into a product of primes

• Each number, if not prime can be written as the product of 2 other numbers

• If these are not prime then they are written as the product of 2 numbers.

• This cannot continue forever

Infinite number of Primes

• Euclid states that Prime numbers are more than any assigned multitude of prime numbers.

• “An infinite number of primes exist”

• proven geometrically

A

B

C

E

G

D

F

A,B, and C are the assigned primes

DE is least number measured by all three

Then EF is either prime, or it is not prime

If prime, there is another prime found

If not prime, then this line must be measured by some prime number G, and this is a new prime.

Why is G different from the others?

• G is not similar to A,B, or C because if it were, then G would measure DE and DF, and it would be absurd for G to measure the segment and it’s remainder.

• (Proof given by John Fauvel)

Euclid got it all started.

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