How do you find the general solution of a Diophantine equation?
Solve the linear Diophantine Equation 20x+16y=500,x,y∈Z+.
- Solution.
- Step 1: gcd(20,16)=4.
- Step 2: A solution is 4125=20(1)(125)+16(−1)(125).
- Step 3: Let u = x – 125 and v = y + 125.
- Step 4: In general, the solution to ax + by = 0 is x=bdk and y=-adk, kZ \ {0}, d=gcd(a,b).
- Step 5: Replace u and v.
What does find the solution to the equation mean?
A solution to an equation is a value of a variable that makes a true statement when substituted into the equation. The process of finding the solution to an equation is called solving the equation. To find the solution to an equation means to find the value of the variable that makes the equation true.
What is the purpose of Diophantine equation?
The purpose of any Diophantine equation is to solve for all the unknowns in the problem. When Diophantus was dealing with 2 or more unknowns, he would try to write all the unknowns in terms of only one of them.
What is Diophantine equation used for?
What does solve mean in mathematics?
To solve something is to find a solution, like figuring out the answer to a complex riddle. The verb solve is often used in mathematics, and it means to answer a math problem.
How do you find solutions in math?
To find the solution set from the replacement set, plug in each value from the replacement set and evaluate both sides of the equation. If the two sides are equal, the equation is true and thus the value is a solution. Example 1: Find the solution set of 11 – 5w = 1 from the replacement set {0, 2, 4}.
What is a Diophantine equation?
A Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integral solutions are required. An Integral solution is a solution such that all the unknown variables take only integer values.
What is the difference between linear and exponential Diophantine?
A linear Diophantine equation equates the sum of two or more monomials, each of degree 1 in one of the variables, to a constant. An exponential Diophantine equation is one in which exponents on terms can be unknowns.
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.
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.