What is the meaning of Diophantine equation?
In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones (an integer solution is such that all the unknowns take integer values).
Is Diophantine equation solvable?
For instance, we know that linear Diophantine equations are solvable.
How do you prove a Diophantine equation?
d=as+bt. dm=(as+bt)mc=a(sm)+b(tm). This means that x=sm, y=tm is a solution of ax+by=c, and we have proved that the Diophantine equation ax+by=c has at least one solution. x=x0+bdk y=y0−adk.
Does every Pell equation have a solution?
(4) Pell’s equation always has nontrivial solutions. The fundamental solution is x y = { [ a 0 , a 1 , … , a k − 1 ] if k is even [ a 0 , a 1 , … , a 2 k − 1 ] if k is odd .
What is Simon’s factoring trick?
Simon’s favorite factoring trick (SFFT), also called completing the rectangle, is a simple but clever factorization of the expressions of the form x y + x n + y m + m n , xy+xn+ym+mn, xy+xn+ym+mn, where x x x and y y y are variables (usually integer variables), and m m m and n n n are integers.
How do you tell if a Diophantine equation has a solution?
Let a, b and c be integers with a≠0 and b≠0, and let d=gcd(a,b).
- If d does not divide c, then the linear Diophantine equation ax+by=c has no solution.
- If d divides c, then the linear Diophantine equation ax+by=c has infinitely many solutions.
Who gave Pell’s equation?
Brahmagupta was an Indian mathematician in the 7 th 7^\text{th} 7th century AD who was one of the first to study Pell’s equation in general.
What are the different types of Diophantine equations?
Chapter 2 presents classical Diophantine equations, includ- ing linear, Pythagorean, higher-degree, and exponential equations, such as Catalan’s. Chapter 3 focuses on Pell-type equations, serving again as an introduction to this special class of quadratic Diophan- tine equations.
What did Diophantus do in Greek algebra?
Diophantus was the first to employ symbols in Greek algebra. He used a symbol (arithmos) for an unknown quantity, as well as symbols for algebraic operations and for powers. Arithmeticais also significant for its results in the theory of numbers, such as the fact that no integer of the form 8n+7 can be written as the sum of three squares.
Where did didiophantus do his work?
Diophantus did his work in the great city of Alexandria. At this time, Alexandria was the center of mathematical learning. The period from 250 bce to 350 ce in Alexandria is known as the Silver Age, also the Later Alexandrian Age. This was a time when mathe- maticians were discovering many ideas that led to our current con- ception of mathematics.
Did Diophantus quote from Hypsicles?
Diophantus did quote the definition of a polygonal number from the work of Hypsicles, who was active before 150 bce,sowecan conclude that Diophantus lived after that date. From the other end, Theon, a mathematician also from Alexandria, quoted the work of Diophantus in 350 ce.
