General solution of Euclidean algorithm and linear Diophantine equation
Dutch mathematician
and historian
Bartel Leendert van der Waerden
suggests that the Euclidean algorithm
originated from a number theory textbook
written by Pythagorean mathematicians.
This algorithm may have been known
by Eudoxus of Cnidus (375 BC).
Euclid and Aristotle also used the term
ἀνθυψαιρεσις (anthyphairesis).
It may have been known before Eudoxus.
ax+by=c (where a, b and c are given integers)
This Diophantine equation has a solution
(where x and y are integers)
only if c is a multiple of
the greatest common divisor of a and b.
To expand the Euclidean algorithm
and find a general solution to the linear Diophantine equation
Let’s take a look at this and briefly look at Bézout’s identity.
The 5th century Indian mathematician
Aryabhata called this algorithm kuṭṭaka (कुट्टक).
I think he was amazed that
all Diophantine equations were crushed or cut.
It was made with a lot of effort over a long period of time.
I hope you enjoy it and find it helpful.
Please see the video for more details.
#Mathematician #Bézout’s identity #Indefinite equation
