Mathematics Hothouse

The name/word Diophantine equation honours the mathematician Diophantus, who initiated the study of such equations. Practically, nothing is known of Diophantus as an individual, save that he lived in Alexandria sometime around 250 A.D. The only positive evidence as to the date of his activity is that the Bishop of Laodicea, who began his episcopate in 270, dedicated a book on Egyptian computation to his friend Diophantus. Although Diophantus’ works were written in Greek and he displayed the Greek genius for theoretical abstraction, he was most likely a Hellenized Babynolian. The only personal particulars we have of his career come from the wording of an epigram-problem (apparently dating from the 4th century). His boyhood lasted 1/6 of his life; his beard grew after 1/12 more; after 1/7 more he married; and his son was born 5 years later, the son lived to half his father’s age and the father died 4 years after his son. If x was the age at which Diophantus died, these data lead to the equation:

with solution . Thus, he must have reached an age of 84, but in what year or even in what century is not certain.

The great work upon which the reputation of Diophantus rests is his Arithmetica, which may be described as the earliest treatise on algebra. Only six Books of the original thirteen have been preserved. It is in the Arithmetica that we find the first systematic use of mathematical notation, although the signs employed are of the nature of abbreviations for words rather than algebraic symbols in the sense with which we use them today. Special symbols are introduced to represent frequently occurring concepts, such as the unknown quantity in an equation  and the different powers of the unknown up to the sixth power. Diophantus also had a symbol to express subtraction, and another for equality.

It is customary to apply the term Diophantine equation to any equation in one or more unknowns that is to be solved in the integers. The simplest type of Diophantine equation is the linear Diophantine equation in two variables:

where a, b, c are given integers and a, b are not both zero. A solution of this equation is a pair of integers that, when substituted in to the equation, satisfy it; that is, we ask that . Curiously enough, the linear equation does not appear in the extant works of Diophantus (the theory required for its solution is to be found in Euclid’s Elements), because he viewed it as trivial, most of his problems deal with finding squares or cubes with certain properties.

Reference:

Elementary Number Theory, David M. Burton, 6th edition, Tata McGraw Hill Edition.

More such interesting information about famous mathematical personalities is found in the classic, “Men of Mathematics by E. T. Bell”.

— Nalin Pithwa