To find two numbers such that their difference and the difference of their cubes are equal to two given numbers. (Write the equations as x−y=a, x3−y3=b. Diophantus takes a=10, b= 2120.) Derive necessary conditions on a and b that ensure a rational solution

To find two numbers \( x \) and \( y \) such that their difference and the difference of their cubes are equal to two given numbers, we are given the system of equations:

\[
x - y = a \tag{1}
\]
\[
x^3 - y^3 = b \tag{2}
\]

### Step-by-step derivation:

1. **Express \( x \) in terms of \( y \):**
   From equation (1), we have:
   \[
   x = y + a \tag{3}
   \]
   Substitute this into equation (2).

2. **Substitute \( x = y + a \) into the cubic equation:**
   We use the identity for the difference of cubes:
   \[
   x^3 - y^3 = (x - y)(x^2 + xy + y^2)
   \]
   Using equation (1) \( x - y = a \), we get:
   \[
   x^3 - y^3 = a \cdot (x^2 + xy + y^2) \tag{4}
   \]
   Now substitute \( x = y + a \) from equation (3) into this expression for \( x^2 + xy + y^2 \).

3. **Expand \( x^2 + xy + y^2 \):**
   Substituting \( x = y + a \) into the quadratic terms:
   \[
   x^2 = (y + a)^2 = y^2 + 2ay + a^2
   \]
   \[
   xy = (y + a)y = y^2 + ay
   \]
   Now sum these up:
   \[
   x^2 + xy + y^2 = (y^2 + 2ay + a^2) + (y^2 + ay) + y^2 = 3y^2 + 3ay + a^2 \tag{5}
   \]

4. **Substitute back into equation (4):**
   Substituting \( x^2 + xy + y^2 = 3y^2 + 3ay + a^2 \) into equation (4), we get:
   \[
   x^3 - y^3 = a \cdot (3y^2 + 3ay + a^2) = b
   \]
   Expanding this:
   \[
   3a y^2 + 3a^2 y + a^3 = b \tag{6}
   \]

### Necessary conditions for rational solutions:

Equation (6) is a quadratic equation in \( y \):
\[
3a y^2 + 3a^2 y + (a^3 - b) = 0 \tag{7}
\]
For \( y \) to have a rational solution, the discriminant of this quadratic equation must be a perfect square (since rational roots of a quadratic require the discriminant to be a perfect square).

The discriminant \( \Delta \) of the quadratic equation \( 3a y^2 + 3a^2 y + (a^3 - b) = 0 \) is given by:
\[
\Delta = (3a^2)^2 - 4 \cdot 3a \cdot (a^3 - b)
\]
Simplifying:
\[
\Delta = 9a^4 - 12a(a^3 - b)
\]
\[
\Delta = 9a^4 - 12a^4 + 12ab
\]
\[
\Delta = -3a^4 + 12ab
\]
Thus, for \( y \) to be rational, the discriminant \( -3a^4 + 12ab \) must be a perfect square.

### Example with \( a = 10 \) and \( b = 2120 \):

For \( a = 10 \) and \( b = 2120 \), we substitute into the discriminant formula:
\[
\Delta = -3(10)^4 + 12(10)(2120)
\]
\[
\Delta = -3(10000) + 12(10)(2120)
\]
\[
\Delta = -30000 + 254400
\]
\[
\Delta = 224400
\]
Since \( 224400 \) is a perfect square (it's equal to \( 474^2 \)), there exists a rational solution in this case.

### Necessary condition:

To ensure a rational solution for \( y \), the discriminant \( \Delta = -3a^4 + 12ab \) must be a perfect square.

---

Would you like to explore any particular step in more detail or need further clarification?

Here are five related questions to expand the discussion:
1. How does the value of \( a \) affect the discriminant?
2. Can we solve for specific values of \( y \) when the discriminant is a perfect square?
3. What happens if \( \Delta \) is not a perfect square?
4. How does this method change if we modify the equation \( x^3 - y^3 = b \)?
5. How can we generalize this approach for different forms of polynomials?

**Tip:** Always check the discriminant when solving quadratic equations for rational solutions!

To find two numbers such that their difference and the difference of their cubes are equal to two given numbers. (Write the equations as x−y=a, x^3−y^3=b. Diophantus takes a=10, b=2120.) Derive necessary conditions on a and b that ensure a rational solution.

Explore how to find two numbers such that their difference and the difference of their cubes equal two given numbers, using the system of equations x - y = a and x^3 - y^3 = b. Learn about the discriminant condition for rational solutions, with a detailed example for a = 10 and b = 2120.

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