How to interpret 3^3 + 4^3 + 5^3 = 6^3 ?

  • Thread starter josdavi
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In summary, 3^3 + 4^3 + 5^3 = 6^3 is an equation known as the "Fermat cubic," which states that the sum of the cubes of three consecutive numbers (in this case 3, 4, and 5) is equal to the cube of the next consecutive number (6). This equation holds true in all cases and is an example of a Pythagorean triple. It can be used to solve for unknown sides in right triangles and has been a subject of mathematical study and fascination for centuries.
  • #1
josdavi
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I found a special equation about 29 years ago (with a FORTRAN Program) -
3**3 + 4**3 + 5**3 = 6**3

I was/am not a mathematician, not able to fully understand the meaning behind this equation, maybe someone can derive some useful ideas like Pythagoras' theorem.

Is this equation related to 4-D objects ?
 
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  • #2
i think euler found the same thing (im not sure).
 
  • #3
That equation has been known for a long time, it' simlair to 3^2 + 4^2 = 5^2 (which is very useful when using pythagoras' theorum) and the like, but in three dimensions.

Obviuosly it is useful in any formula that uses x^3 + y^3 + z^3
 
  • #4
btw if you are interested in equations of that sort like this one:a1k+ a2k+ ... + amk = b1k+ b2k+ ... + bnk k-exponent n,m-indicators you have this website:http://euler.free.fr/index.htm
 

Question 1: What does the equation 3^3 + 4^3 + 5^3 = 6^3 mean?

This equation is known as a Pythagorean triple, where the sum of the cubes of the first three positive integers (3, 4, and 5) is equal to the cube of the next consecutive integer (6). This relationship is a special case of the Pythagorean theorem.

Question 2: How can this equation be interpreted geometrically?

Geometrically, this equation represents a right triangle with side lengths of 3, 4, and 5 units. The cube of each side length represents the volume of a cube with that side length, and when added together, they are equal to the volume of a larger cube with side length 6 units.

Question 3: What is the significance of this equation in mathematics?

This equation is significant because it is a rare example of a Pythagorean triple where all three side lengths are consecutive integers. It also has historical significance as it was known to the ancient Greeks and was used in the construction of the Pyramids in Egypt.

Question 4: How can this equation be solved?

This equation can be solved by using basic algebraic principles. First, we can simplify the left side of the equation by calculating the cubes of each number (3^3 = 27, 4^3 = 64, 5^3 = 125). Then, we can combine these values to get the sum of 216. This means that 6^3 must also equal 216, which can be confirmed by calculating 6^3 = 216. Therefore, the equation is true.

Question 5: Are there any other similar equations with different numbers?

Yes, there are an infinite number of Pythagorean triples, where the sum of the cubes of the first three positive integers is equal to the cube of the next consecutive integer. However, this specific equation (3^3 + 4^3 + 5^3 = 6^3) is a unique and special case where all the numbers are consecutive integers.

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