Curious Number Theory Problem

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Homework Statement :
Recently, a group of fellow math nerds and myself stumbled upon an interesting problem. The problem is stated: "Find the average number of representations of a positive integer as the sum of two squares."

Relevant equations:
N = a^(2) + b^(2), where a and b can be 1 or many corresponding sets. But we are looking for the average number of sets, for all real numbers of N.

The attempt at a solution:

Clearly, I realize a limit will need to be taken, somehow, of all numbers N that can be expressed as the sum of two squares. Naturally, the general equation for N would be N = a^(2) + b^(2). Also, I figured that N must meet the condition that every prime number of the form 4k+3 appears an even number of times in it's prime factorization. I have no idea how to connect these two conditions, or if more must be met. If it's any help, my friend claimed that the answer is π, but had no way to prove it.

Any ideas are greatly appreciated!
 

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  • #2
SammyS
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Homework Statement :
Recently, a group of fellow math nerds and myself stumbled upon an interesting problem. The problem is stated: "Find the average number of representations of a positive integer as the sum of two squares."

Relevant equations:
N = a^(2) + b^(2), where a and b can be 1 or many corresponding sets. But we are looking for the average number of sets, for all real numbers of N.

The attempt at a solution:

Clearly, I realize a limit will need to be taken, somehow, of all numbers N that can be expressed as the sum of two squares. Naturally, the general equation for N would be N = a^(2) + b^(2). Also, I figured that N must meet the condition that every prime number of the form 4k+3 appears an even number of times in it's prime factorization. I have no idea how to connect these two conditions, or if more must be met. If it's any help, my friend claimed that the answer is π, but had no way to prove it.

Any ideas are greatly appreciated!
I'm a bit confused by your write-up.

Are you find the average number of such representations over the set of real numbers or over the set of positive integers (the set of natural numbers)? ... or does it not matter?
 
  • #3
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If a and b are real numbers then there are an infinite number of representations. So they must be integers, natural numbers, or counting numbers.

Then if N is a real number it almost never has a representation with whole numbers. So it's got to be a whole number too. But then there is usually no representation. So I don't see how pi can be the answer. It is either infinite or close to zero, I think.
 
  • #4
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I'm a bit confused by your write-up.

Are you find the average number of such representations over the set of real numbers or over the set of positive integers (the set of natural numbers)? ... or does it not matter?
Unfortunately, that one sentence stating the problem is the only information given. My guess is it probably means the natural numbers.
 

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