Proving the Unsolvable: Lagrange's Theorem and 4 Squares

In summary, a question was posed about proving that at least one of any four consecutive natural numbers is not a sum of two squares. A hint was given to consider four consecutive numbers and their squares mod 4, leading to a contradiction. The person asking for help thanked for the hint and confirmed understanding.
  • #1
RichardCypher
14
0
Hi everybody :smile:
I'm currently reading Burton's Elementary Number Theory (almost done!) and in the chapter about Lagrange's Theorem about the sum of four squares, there is a supposedly easy question which I can't solve for some reason :blushing:. I'd really appreciate a hint or two...

Prove that at least one of any four consecutive natural numbers is not a sum of two squares [that is, can't be represented as the sum of two squares of whole numbers]

Thank you all! :smile:
 
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  • #2
Well I think this one works?

2 + 3 + 4 + 5 = 14

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16

It has to be the SUM of TWO squares... 9 + 4 = 13. None of the others work.
 
  • #3
Consider four consecutive numbers mod 4, then consider squares mod 4. The result follows immediately.

iwin2000: the problem was to show the result for all {n, n + 1, n + 2, n + 3}, not just for one such instance.
 
  • #4
CRGreathouse said:
Consider four consecutive numbers mod 4, then consider squares mod 4. The result follows immediately.

iwin2000: the problem was to show the result for all {n, n + 1, n + 2, n + 3}, not just for one such instance.

The square of any natural number mod 4 has to be 0 or 1. Therefore, the sum of two such squares mod 4 has to be 0, 1, or 2. However, out of four consecutive natural numbers mod 4, one has to be 3. Contradiction. Is that right?

Great hint! Thank-you very much :biggrin:
 
  • #5
You got it.

I like minimal hints.
 

Related to Proving the Unsolvable: Lagrange's Theorem and 4 Squares

1. What is Lagrange's Theorem?

Lagrange's Theorem, also known as the Four Squares Theorem, states that every positive integer can be represented as the sum of four squares. This means that any positive integer can be written as a^2 + b^2 + c^2 + d^2, where a, b, c, and d are all integers.

2. Why is it called "unsolvable" if there is a theorem for it?

The term "unsolvable" in this context refers to the fact that there is no general formula or algorithm for finding the specific values of a, b, c, and d that satisfy the theorem for a given number. While the theorem proves that such values exist, it does not provide a method for finding them.

3. Can Lagrange's Theorem be applied to all numbers?

Yes, Lagrange's Theorem applies to all positive integers. However, it does not apply to negative numbers or fractions.

4. How was Lagrange's Theorem proven?

Lagrange's Theorem was proven by the mathematician Joseph-Louis Lagrange in 1770. He used mathematical techniques such as modular arithmetic and number theory to prove the theorem.

5. Are there any practical applications of Lagrange's Theorem?

While Lagrange's Theorem may seem like a purely theoretical concept, it has practical applications in fields such as cryptography and computer science. It is also used in number theory to study properties of integers and their relationships with other mathematical concepts.

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