April 21, 2017 Friday
Bedtime Story
Proof of Euclid's Theorem
Euclid’s Theorem goes something like this if not exactly like the
way I write below.
Let us consider a finite list of prime number p1, p2,…, pn
It will be shown that yet another prime number exists that is not
contained in the given list.
Let P be the product of all the listed primes.
So P = p1p2p3…pn
Let q = P + 1
Then q can either be a prime or not be prime.
If q turns out to be a prime, then we have a new prime that was
not on the original proposed list.
On the other hand, if q turns out not be a prime number, then
there has to be some prime factor p like p1, p2, p3…that divides q.
Prime factors are nothing but prime numbers that divide the
integer exactly.
The prime factors for 120 are:
2 x 2 x 2 x 3 x 5 or 23 x 3 x 5
If this factor p that divides q was already on our old list, then
it would also divide P, the product of all the listed primes.
This should be so as the P is product of all the listed primes.
But p divides q which is P + 1.
If this prime p divides both P and q, then it would have to divide
the difference of P and q which is (P + 1) – P or 1.
Since no prime number divides 1 (1 is not considered a prime
number), the prime p cannot be on the original list.
This proves there exists at least one more prime number beyond
those in the list.
The logic used in this proof is known as reductio ad absurdum, which
essentially is a technique that begins with premise and ends with the
conclusion of it not being possible as it would then lead to impossible
outcomes.
The point that I am about to make for this proof is not very
obvious.
At first glance, the proof looks to be complete with all the chain
of reasoning listed.
Yet if probed deeper, certain rules of inference and theorems of
logic have been surreptitiously invoked.
They are widely accepted yet the fact remains that they have not
been explicitly stated here.
They belong to elementary logic either very basic or advanced.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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