Diophantus

Something is always better than nothing.
Well, i tried to solve Koushik's Quest..., concluded pretty soon that it was not going to be that easy and then decided to "ask-my-friend".(i.e. google search)
My search was not up to the mark this time, but i ended up going through a lot of material on Diophantine equations.

Here is koushik's question :

Can you solve this problem ?
For given prime number p, solve the equation in set of integers Z:
xyz=p(x+y+z)

Koushik, did you manage to solve this ?

Back to Diophantus.
Greek Mathematician. Lived from about 200 - 284 A.D in Alexandria, Egypt. Historians could find his age only from one of his puzzles. (a classic - if my father's age is twice that of blah..blah..).
He wrote Arithmetica a collection of over 130 problems and their solutions. The class of equations that he dealt with most came to be known as Diophantine equations. A Diophantine equation is a polynomial equation with integral coefficients to which only integral solutions are sought. The study of Diophantine equations is one of the central areas of number theory. Fermat's Last Theorem is one such equation.



That's the front page of his book. (ofcourse printed, republished,

Diophantus may not be, as he is often called, the father of Algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.
This considering the fact that symbolism (usage of short notations) was not prevalent those days.

Instead of "x+y" he would say "the first unknown added to the second unknown".
For "(12+6n)/(n^2-3)" he says "a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three".
SK's chemistry book was far better.

We should be able to appreciate the advantage of symbolism, and also take pride in the fact that Indians were the first to employ a lot of such techniques. ( ahh.. think short-cuts think Indian ! )
The Arithmetica remained the authority in Algebra for many more years to come. It is on one of the margins of the pages on one of these that Fermat made the famous (some call it sadistical) remark : " I have a truly marvelous proof of this (his last theorem) which this margin is too small to contain".

One Joke for today :

A physicist, an Astronomer and a Mathematician are travelling on a train in Scotland.
Suddenly the Astronomer peeps out of the window and exclaims
"Look at the sheep out there ! Looks like sheep in Scotland are black !"
The Physicist says "Correction.. Some sheep in Scotland are black"
The Mathematician finally says "This only means.."
"..that in scotland there exists atleast one field with atleast one sheep, atleast one side of which is black".