# The Sum of Cubes Problem: What’s the answer to x3+y3+z3=k?

## Introduction

For decades, a math puzzle has stumped the smartest mathematicians in the world. The equation x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that’s sometimes known as “summing of three cubes.”

The sum of cubes problem is a specific type of Diophantine equation, which is an equation that requires integer solutions. The problem has fascinated mathematicians for centuries, and despite its seemingly simple nature, it has proven to be extraordinarily difficult to solve.

In this article, we’ll explore the history of the sum of cubes problem, its significance in mathematics, and the breakthrough solution that was discovered in 2019.

## FAQs

### What is the sum of cubes problem?

The sum of cubes problem, also known as the Diophantine equation x3+y3+z3=k, asks whether it is possible to find three whole numbers that can be added together to make another whole number when raised to the third power. For example, 1³ + 12³ + (-13)³ = 1. The problem asks whether this is possible for all numbers k from 1 to 100.

### When was the sum of cubes problem first introduced?

The sum of cubes problem has been around for centuries. The ancient Indian mathematician Ramanujan wrote about it in the early 1900s, and the problem was brought to the attention of Western mathematicians in the 1950s. Since then, it has been a popular topic of research.

### Why is the sum of cubes problem so difficult to solve?

The sum of cubes problem is difficult to solve for several reasons. Firstly, it involves finding integer solutions to a Diophantine equation, which is a notoriously difficult type of problem. Secondly, there are a large number of possible values for k, which makes it difficult to find a general solution that works for all values. Finally, the problem requires finding solutions that involve combining negative and positive numbers in specific ways, which adds another layer of complexity to the problem.

### What was the solution to the sum of cubes problem?

In 2019, mathematicians Andrew Booker and Andrew Sutherland discovered a solution to the sum of cubes problem. They found that 42, 65, and 90 can be written as the sum of three cubes:

42 = (-80538738812075974)3 + 80435758145817515³ + 12602123297335631³

65 = (-569936821221962380720³) + (-569936821113563493509³) + 472715493453327032³

90 = (-149389005988152413840³) + 158960462321162570³ + 69194705831948245³

Notably, this discovery disproved the popular belief that 33 was impossible to write as the sum of three cubes. With the use of a supercomputer, the mathematicians were able to exhaustively search for solutions to the problem, paving the way for further research and discoveries in the future.

### What is the significance of the sum of cubes problem?

Like many problems in mathematics, the significance of the sum of cubes problem lies in its ability to expand our understanding of number theory and algebra. The problem has fascinated mathematicians for decades, and its solution represents a significant breakthrough in the field. Additionally, the use of supercomputers to search for solutions may lead to advancements in computer science and technology as well.

## Conclusion

The sum of cubes problem is just one example of the fascinating and complex problems that mathematicians grapple with every day. The recent breakthrough in finding a solution to the problem represents a significant step forward in number theory and algebra, and it will no doubt inspire new research and discoveries in the coming years.

If you’re interested in learning more about the sum of cubes problem or other mathematical puzzles, there are plenty of resources available online, including mathematical forums, online courses, and research papers. By exploring the world of math, you may just discover a new passion or a creative solution to an old problem.

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