Lecture 19We continued looking at Book VII,
Book VIII and
Book IX.
To be precise we started with the Euclidean Division Algorithm and how to compute the greatest common divisor
of two given numbers. Read carefully the commentaries of:
We computed carefully computed the gcd of 3796 and 1387. Well....it turns out that it is 73. We used
the Euclidian Algorithm to find it, and to prove that it is a divisor of both numbers and that actually
it is the greatest divisor. It is actually possible to write the gcd as a combination of the two given numbers.
That is, we can find numbers a and b such that
a*3796 + b*1387.
a = ?
... and ... b = ?? ... ;-)We then moved on and we commented the following propositions: (we proved Proposition 20...in a modern language, though!) Finally, we arrived (in Book IX) to Proposition 14, which Victor Katz reads as a version of the Fundamental Theorem of Arithmetic, that is "every number
can be uniquely factored into product of primes."
We then ended the lecture with Euclid's famous proof about the infinitude of prime numbers: Prime numbers are more than any assigned multitude of prime numbers, Comment:
Prime numbers are at the base of our encription systems for all secure transactions that we do over
the Internet. Do you know for instance how RSA works? If not, this is a good idea for a topic for your
final project. It boils down to Fermat's Little Theorem.
Click on the above links to go directly to the proofs and commentaries of the results. |