Exploring Integer Solutions to a^3+b^3+c^3=d^3

  • A
  • Thread starter kuruman
  • Start date
  • Tags
    Integer
In summary, there is a remarkable equation ##3^3+4^3+5^3=6^3## that forms a "cubic quartet" and extends Fermat's last theorem. There are likely more solutions and there are various formulas to generate such equations.
  • #1
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2023 Award
14,359
7,838
I am wondering about the integer solutions to ##a^3+b^3+c^3=d^3~##.

By trial and error I stumbled upon ##3^3+4^3+5^3=6^3##. I find this equation remarkable in that not only the four integers are consecutive, but also because the three integers on the left form the well known Pythagorean triplet, ##3^2+4^2=5^2##.

It's like an extension of Fermat's last theorem to what might be called "cubic quartets" by analogy to Pythagorean triplets. If there is one solution, there are probably more. How does one find them other than multiplying an already known solution by a common factor? Has anyone looked into this?
 
Mathematics news on Phys.org
  • #2
This wikipedia page about sums of powers has various formulas (or references to formula) to generate equations like this.

As an example, for all integers a,b,
$$a^3(a^3+b^3)^3 = b^3(a^3+b^3)^3\,+\,a^3(a^3-2b^3)^3\,+\,b^3(2a^3-b^3)^3$$
 
  • Like
Likes kuruman
  • #3
Ah, yes. Thank you, I didn't know how to look for it.
 

1. What is the significance of exploring integer solutions to a^3+b^3+c^3=d^3?

The equation a^3+b^3+c^3=d^3 is known as the "sum of three cubes" and has been a topic of interest in mathematics for centuries. Exploring integer solutions to this equation can help us understand the properties of numbers and their relationships, as well as provide insights into other areas of mathematics such as number theory and algebra.

2. Are there any known integer solutions to a^3+b^3+c^3=d^3?

Yes, there are several known integer solutions to this equation. The most famous is the solution known as "Fermat's Last Theorem," which states that there are no integer solutions for a, b, c, and d when n is greater than 2. However, there are infinitely many solutions when n=2, which have been discovered and studied by mathematicians.

3. How do scientists explore integer solutions to a^3+b^3+c^3=d^3?

Scientists use a variety of methods to explore integer solutions to this equation. These methods include algebraic techniques, number theory, and computer simulations. By using these methods, scientists can analyze the properties of numbers and look for patterns and relationships that can help them find integer solutions to a^3+b^3+c^3=d^3.

4. What are some applications of exploring integer solutions to a^3+b^3+c^3=d^3?

Exploring integer solutions to this equation has many practical applications, including cryptography, coding theory, and cryptography. For example, the study of integer solutions can help us develop more secure encryption methods and error-correcting codes.

5. How does the study of integer solutions to a^3+b^3+c^3=d^3 contribute to our knowledge of mathematics?

The study of integer solutions to this equation contributes to our knowledge of mathematics in several ways. It helps us understand the properties of numbers and their relationships, provides insights into other areas of mathematics, and can lead to the development of new mathematical theories and concepts. Additionally, the study of integer solutions can also inspire new research and advancements in mathematics.

Similar threads

MHB Positive Integer Triples $(a,b,c)$ Satisfying $a^3+b^3+c^3=(abc)^2$
    • General Math
Replies
1
Views
928
MHB Solve for $d-b$: $a^5=b^4,\,c^3=d^2,\,c-a=19$
    • General Math
Replies
1
Views
765
MHB V6 a*b=9; v7 a*b=a^3+b^3
    • General Math
Replies
1
Views
550
MHB Find Solutions to a+b+c+d=4, a^2+b^2+c^2+d^2=6, a^3+b^3+c^3+d^3=94/9 in [0,2]
    • General Math
Replies
2
Views
745
MHB Positive Integers: Evaluate $a+b+c$
    • General Math
Replies
1
Views
722
MHB Prove No Integers Solve $ax^3+bx^2+cx+d=1$ for x=19,2 for x=62
    • General Math
Replies
3
Views
846
MHB Show that for integer a,b,c the product abc(a^3−b^3)(b^3−c^3)(c^3−a^3) is divisible by 7
    • General Math
Replies
1
Views
890
MHB J1.1.6 Suppose a and b are integers that divide the integer c
    • General Math
Replies
7
Views
3K
A Differences between Gaussian integers with norm 25
    • General Math
Replies
2
Views
1K
B On the representation of integers?
    • General Math
Replies
16
Views
2K

Hot Threads

Recent Insights

Back
Top