Generalized version of cannon ball problem

In summary, the conversation discusses the possibility of finding a solution for the equation where for all p in the natural numbers, there exists an n greater than 1 where the sum of k^p from k = 1 to n equals C^2, where C is an arbitrary natural number. The only known solution is n = 24, p = 2, and C = 70, but the question is whether there are other solutions for arbitrary p. The conversation mentions the difficulty of proving the existence of solutions for arbitrary p using congruence conditions, and suggests looking into Diophantine equations for more information.
  • #1
kevin0960
12
0
For All p in Natural Number,
Is [tex]\exists n , n > 1, \sum^{n}_{k=1} k^p = C^2 [/tex] where C is arbitary natural number (not constant) ??
 
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  • #2
kevin0960 said:
For All p in Natural Number,
Is [tex]\exists n , n > 1, \sum^{n}_{k=1} k^p = C^2 [/tex] where C is arbitary natural number (not constant) ??

As far as I'm aware the only solution is:
n = 24
p = 2
C = 70
 
  • #3
BruceG said:
As far as I'm aware the only solution is:
n = 24
p = 2
C = 70

No, I checked with mathematica n < 100,000, p < 20

there are some solutions such as

p = 5,
n = 13, 134, etc

I think there are more solutions.. :)
 
  • #4
You're trying to solve a sequence of Diophantine equations. For a famous case, search for square triangular number.
 
  • #5
CRGreathouse said:
You're trying to solve a sequence of Diophantine equations. For a famous case, search for square triangular number.

Yeah, I know

But what I mean was, is it possible to find the solution for arbitary p ??
 
  • #6
kevin0960 said:
But what I mean was, is it possible to find the solution for arbitary p ?

Probably not. Diophantine equations are hard, in the sense of the negative answer to Hilbert's 10th.

But for any given p it should be possible to at least formulate the problem in that form to see if anything can be discovered. So, for example, with p = 7 you have

[tex]3x^8 + 12x^7 + 14x^6 - 7x^4 + 2x^2=24y^2[/tex]
 
  • #7
CRGreathouse said:
Probably not. Diophantine equations are hard, in the sense of the negative answer to Hilbert's 10th.

But for any given p it should be possible to at least formulate the problem in that form to see if anything can be discovered. So, for example, with p = 7 you have

[tex]3x^8 + 12x^7 + 14x^6 - 7x^4 + 2x^2=24y^2[/tex]

It seems like we cannot find C for arbitary p,

But can we know the existence of C for arbitary p ??

I don't need to find the entire solutions, just a single one.
 
  • #8
It's not clear that a solution exists for a given p. If not, I don't know of an easy way to prove it -- congruence conditions won't be enough, since n = 1 works and so there are always good congruence classes mod any prime power.
 
  • #9
CRGreathouse said:
It's not clear that a solution exists for a given p. If not, I don't know of an easy way to prove it -- congruence conditions won't be enough, since n = 1 works and so there are always good congruence classes mod any prime power.

Yeah, It looks almost impossible to use modular to prove...

Do you know any related article about this??
 

1. What is the Generalized Version of Cannon Ball Problem?

The Generalized Version of Cannon Ball Problem is a mathematical problem that involves finding the number of ways to arrange a set number of objects into a pyramid shape, with each consecutive layer containing one more object than the previous layer. It is similar to the classic cannon ball problem, where the goal is to find the number of cannon balls that can fit in a pyramid shape with a given number of layers.

2. How is the Generalized Version of Cannon Ball Problem solved?

The Generalized Version of Cannon Ball Problem can be solved using a mathematical formula known as the Gaussian sum. This formula involves adding up the number of objects in each layer of the pyramid, starting from the bottom layer and working up to the top. The result is then divided by 2 to account for the symmetry of the pyramid.

3. What is the significance of the Generalized Version of Cannon Ball Problem?

The Generalized Version of Cannon Ball Problem has been used in various fields, including mathematics, physics, and computer science. It is a classic example of a problem that involves finding patterns and using mathematical formulas to solve it. It also has real-world applications, such as calculating the number of ways to stack objects or arrange items in a given space.

4. Are there any variations of the Generalized Version of Cannon Ball Problem?

Yes, there are several variations of the Generalized Version of Cannon Ball Problem. Some variations include using different shapes, such as cubes or spheres, instead of cannon balls. There are also variations where the pyramid is built in different orientations or with different constraints, such as a limited number of objects per layer.

5. Can the Generalized Version of Cannon Ball Problem be solved for any number of layers?

Yes, the Generalized Version of Cannon Ball Problem can be solved for any number of layers. The formula for solving it is applicable to any number of layers, as long as the pyramid can be constructed with that number of objects. However, the number of possible arrangements may become too large for practical calculations for very large numbers of layers.

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