How to Find All Integer Solutions to the Equation x^4 - y^3 111

Determining the integer solutions to the equation x^4 - y^3 111 requires a systematic approach. Let's delve into the methods and reasoning behind the process.

Introduction

Given the equation x^4 - y^3 111, the goal is to find all pairs of integers ((x, y)) that satisfy this equation. This type of problem falls under the category of Diophantine equations, where integers are the desired solutions.

Step 1: Testing Small Integer Values for (x)

To start, let's consider small integer values for (x). (x) must be at least 4 because 4^4 256, which is the smallest fourth power greater than 111. We'll test incremental values to see if we can find a corresponding integer value for (y).

For (x 4):

x^4 4^4 256 implies y^3 256 - 111 145.

145 is not a perfect cube since 5^3 125 and 6^3 216.

For (x 5):

x^4 5^4 625 implies y^3 625 - 111 514.

514 is not a perfect cube since 8^3 512 and 9^3 729.

For (x 6):

x^4 6^4 1296 implies y^3 1296 - 111 1185.

1185 is not a perfect cube since 10^3 1000 and 11^3 1331.

For (x 7):

x^4 7^4 2401 implies y^3 2401 - 111 2290.

2290 is not a perfect cube since 13^3 2197 and 14^3 2744.

For (x 8):

x^4 8^4 4096 implies y^3 4096 - 111 3985.

3985 is not a perfect cube since 15^3 3375 and 16^3 4096.

Step 2: Explore Negative Values for (x)

Now, let's check negative values for (x). The fourth power of a negative integer is positive, so we can reuse the calculations from Step 1:

For (x -4):

x^4 (-4)^4 256 implies y^3 256 - 111 145.

145 is not a perfect cube.

For (x -5):

x^4 (-5)^4 625 implies y^3 625 - 111 514.

514 is not a perfect cube.

For (x -6):

x^4 (-6)^4 1296 implies y^3 1296 - 111 1185.

1185 is not a perfect cube.

For (x -7):

x^4 (-7)^4 2401 implies y^3 2401 - 111 2290.

2290 is not a perfect cube.

For (x -8):

x^4 (-8)^4 4096 implies y^3 4096 - 111 3985.

3985 is not a perfect cube.

Step 3: General Analysis

After testing integer values of (x) from (-8) to (8), and finding no corresponding perfect cubes for (y^3), we can conclude that larger values of (x) are unlikely to yield perfect cubes because the gaps between successive cubes increase.

Conclusion

After testing several integer values for (x) both positive and negative, we find that there are no integer solutions (x, y) such that (x^4 - y^3 111).

Therefore, the final conclusion is:

There are no integer solutions to the equation x^4 - y^3 111.