Curious Number Theory Problem

In summary, curious numbers are positive integers whose digits, when squared and added together, equal the original number. The smallest curious number is 1, and there are other curious numbers such as 25, 39, and 58. There is no limit to how large a curious number can be, but the likelihood of finding one decreases as the number of digits increases. Curious numbers are significant in mathematics due to their unique properties and connections to other areas of math.
  • #1
Ryan888
4
0
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
Ryan888 said:
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
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
SammyS said:
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.
 

What is a curious number?

A curious number is a positive integer that has the property that when its digits are squared and added together, the resulting sum is equal to the original number.

What is the smallest curious number?

The smallest curious number is 1, as 1 squared is 1 and 1+1=1.

Are there any other curious numbers besides 1?

Yes, there are several other curious numbers such as 25, 39, and 58.

Is there a limit to how large a curious number can be?

There is no known limit to how large a curious number can be. However, as the number of digits increases, the likelihood of finding a curious number decreases.

What is the significance of curious numbers in mathematics?

Curious numbers are interesting to mathematicians because they demonstrate a unique property that relates a number to its digits. They also have connections to other areas of mathematics, such as perfect numbers and digital roots.

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