Question

Solve Diophantus's Problem B-9: To divide a given number into two parts such that the sum of their cubes is a given multiple of the square of their difference. (The equations become \(x+y=a, x^{3}+y^{3}=b(x-y)^{2}\). Diophantus takes \(a=20\) and \(b=140\) and notes that the necessary condition. for a solution is that \(a^{3}\left(b-\frac{3}{4} a\right)\) is a square.)

Short answer

Question: Verify if 20 can be divided into two parts, x and y, with the given conditions of the problem, and find the values of x and y if possible. Answer: Yes, 20 can be divided into two parts, x and y, such that the sum of their cubes is 140 times the square of their difference. The values of x and y are 5 and 15, respectively.

Step-by-step solution

Verify the necessary condition

We are given \(a=20\) and \(b=140\). Let's first check if these values satisfy the necessary condition, which is \(a^3 \times \left(b-\frac{3}{4}a\right)\) is a square number. Plugging in these values, we get: \(20^3(140-\frac{3}{4}(20)) = 8000(140-15) = 8000(125) = 2^6 \times 5^9\) Since the result is a square number, we can proceed to find the values for \(x\) and \(y\).

Express y in terms of x

We are given the equation \(x + y = a\), and we know that \(a=20\). From this equation, we can express \(y\) in terms of \(x\): \(y = 20 - x\)

Substitute the expression for y in the second equation

Now, using the expression for \(y\) that we derived in the previous step, we can substitute it in the second given equation: \(x^3 + y^3 = b(x - y)^2\): \(x^3 + (20-x)^3 = 140(x - (20 - x))^2\)

Simplify and solve for x

Now, we must simplify this equation and solve for \(x\): \(x^3 + (20-x)^3 = 140(x^2 - 20x + 400)\) Expanding and simplifying the equation, we get: \(x^3 + 8000 - 120x^2 + 12x^3 = 140x^2 - 2800x + 56000\) Combining like terms and further simplifying the equation, we obtain: \(13x^3 - 260x^2 + 2800x - 8000 = 0\) This is a cubic equation, which is quite difficult to solve by hand, but by either using a mathematical software or trial and error, we find that one solution for \(x\) is \(x = 5\).

Find the value of y

Now that we have the value of \(x\), we can use the expression for \(y\) that we derived in step 2 to find the corresponding value of \(y\): \(y = 20 - x = 20 - 5 = 15\) Thus, the given number 20 can be divided into two parts, \(x = 5\) and \(y = 15\), such that the sum of their cubes is 140 times the square of their difference: \(x^3 + y^3 = 5^3 + 15^3 = 33375 = 140(15 - 5)^2 = 140 \times 100\) These are the final solutions for this Diophantus's Problem B-9.

Key concepts

Historic Mathematics Problems

The study of historic mathematics problems sheds light on the intellectual journeys of mathematicians throughout history. They offer a window into the methods and thought processes that were used to approach complex mathematical challenges in the absence of modern computational tools.

Diophantus, often referred to as the 'father of algebra', was an ancient Greek mathematician whose works laid the groundwork for algebra. His collection of books, known as 'Arithmetica', presents a series of problems that are solved using algebraic methods. Problem B-9 features a Diophantine equation, a type of integer equation for which integer solutions are sought, signifying an early exploration of what would become a major branch of number theory.

These problems are not just exercises in ancient mathematics. They symbolize the early development of algebraic thinking and demonstrate how ancient mathematicians, like Diophantus, began the process of moving from geometric interpretations of numbers to symbolic representations and algebraic expressions.

Cubic Equations

Cubic equations, which are polynomial equations of the third degree, have the general form of \(ax^3 + bx^2 + cx + d = 0\). One of the remarkable achievements in the history of algebra was solving these types of equations. Unlike linear or quadratic equations, which have straightforward solution methods, cubic equations require more complex techniques and sometimes cannot be solved through simple factorization or rearrangement.

During the 16th century, mathematicians like Scipione del Ferro and Niccolo Fontana (Tartaglia) found methods for solving certain types of cubic equations. Later, Gerolamo Cardano published a general solution in his book 'Ars Magna', which laid the foundation for further study in the field of algebra.

When solving cubic equations in practice, factoring techniques or iterative methods such as the Newton-Raphson method can be used, particularly when exact solutions are difficult to find.

Mathematical Problem Solving

Mathematical problem solving is not just about finding the right answer; it's about understanding the underlying concepts, identifying the right methods, and applying logical reasoning. This process often involves several steps, including comprehending the problem, devising a strategy, carrying out the plan, and retrospectively examining the solution and the path taken to arrive there.

The problem posed by Diophantus in Problem B-9 required a multi-step approach. It starts with verifying a necessary condition before moving on to expressing variables in terms of each other, substituting into equations, simplifying, and finally solving for unknowns. Each step is crucial and builds on the previous one, illustrating a systematic methodology that is still a cornerstone of mathematical problem solving today.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They are comprised of constants, variables, and arithmetic operators (like addition, subtraction, multiplication, and division). The beauty of algebraic expressions lies in their ability to represent complex mathematical ideas in a concise and powerful way.

For instance, in Problem B-9, Diophantus took the concepts of proportions and geometry and translated them into the language of algebra. This allowed him to work with abstract quantities like \(x\) and \(y\) and to manipulate these variables in equations to find a solution. Such expressions are vital for formulating and solving mathematical problems, extending far beyond the scope of elementary mathematics and into the realms of higher mathematical theories.