Prove the sum of squares of two odd integers can't be a perfect square

In summary, the conversation discusses a proof regarding the assumption that two odd numbers squared can be perfect squares. The proof uses the equation x^2+y^2=z^2 and assumes x=2j+1 and y=2k+1. It then goes on to produce a contradiction by showing that the sum of the squares of two odd integers is congruent to 2 modulo 4, which contradicts the fact that k^2 can only give remainder 0 or 1 modulo 4. The speaker requests for a detailed explanation of the rest of the proof.
  • #1
kolley
17
0

Homework Statement



x^2+y^2=z^2

Homework Equations





The Attempt at a Solution



assume to the contrary that two odd numbers squared can be perfect squares. Then,
x=2j+1 y=2k+1

(2j+1)^2 +(2k+1)^2=z^2
4j^2 +4j+1+4k^2+4k+1
=4j^2+4k^2+4j+4k+2=z^2
=2[2(j^2+K^2+j+k)+1)]=2s
the book goes on to produce a contradiction having to do with whether s is odd or even. Can someone please walk me through the rest of this proof to the end in detail, because the explanation in my book is very poor. Thank you.
How do I go on to show that this is a contradiction. The book says
 
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  • #2
Note that the sum of the square of two odd integers (2m+1) and (2n+1) is X=4(n^2+m^2+n+m)+2. Clearly, X is congruent to 2 modulo 4. But k^2 can only give remainder 0 or 1 modulo 4.
 

1. What does it mean for a number to be a perfect square?

A perfect square is a number that is the product of two equal integers. In other words, it is the result of squaring an integer.

2. How do you prove that the sum of squares of two odd integers can't be a perfect square?

To prove this, we can use a proof by contradiction. We assume that the sum of squares of two odd integers is a perfect square and then show that this leads to a contradiction. This will prove that our initial assumption was false and therefore, the sum of squares of two odd integers cannot be a perfect square.

3. Can you give an example to demonstrate this statement?

Yes, for example, let's take two odd integers, 3 and 5. The sum of their squares is 9 + 25 = 34. We know that 34 is not a perfect square because the square root of 34 is a decimal number. Therefore, the statement holds true.

4. What is the significance of the integers being odd in this statement?

The significance of the integers being odd is that when we square an odd integer, the result is always odd. This means that when we add two odd squares, the result will always be even. However, a perfect square cannot be even, which is why the sum of squares of two odd integers cannot be a perfect square.

5. Can this statement be generalized to all even integers?

No, this statement only applies to odd integers. This is because when we square an even integer, the result is always even. So, the sum of squares of two even integers can be a perfect square. For example, 4 and 6 are even integers and the sum of their squares, 16 + 36 = 52, is a perfect square.

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