Prove the sum of two even perfect squares is not a perfect square

• vinnie
In summary, the conversation discusses a conjecture that states that for all natural numbers a and b, if a and b are both even, then the sum of their squares is not a perfect square. Different approaches to proving this conjecture are discussed, including proving by contradiction and using counterexamples. Ultimately, it is concluded that the conjecture is false and counterexamples such as 6 and 8 can be used to disprove it.
vinnie

Homework Statement

For all natural numbers, a and b, if a and b are both even, then (a^2+b^2) is not a perfect square. (prove this)

The Attempt at a Solution

I tried proving by contradiction and got (2s)^2 +(2t)^2 =k^2.
which translates to 4s^2 +4t^2=k^2.
I don't know how to form the contradiction from here. Is it even possible?

$$(a+b)^2=a^2+2ab+b^2$$

vinnie said:

Homework Statement

For all natural numbers, a and b, if a and b are both even, then (a^2+b^2) is not a perfect square. (prove this)
Are you sure you have the problem correct as stated? As stated this is easily proven false by counterexample.

D H said:
Are you sure you have the problem correct as stated? As stated this is easily proven false by counterexample.

so I set up the negation, then we assume a and b are even and that a^2 +b^2 is a perfect square. Then subbing 4 for a and 6 for b, we get a contradiction?

Obviously 52 is not a perfect square. The conjecture does not say that the sum of squares of some specific pair of even numbers is not a square number. The conjecture says that the sum of squares of every pair of even numbers is not a square number.

D H said:
Obviously 52 is not a perfect square. The conjecture does not say that the sum of squares of some specific pair of even numbers is not a square number. The conjecture says that the sum of squares of every pair of even numbers is not a square number.

true, but using four and six as counterexamples...

42+62=16+36=52. and 52 is not a perfect square. 4 and 6 do not form a counterexample.

D H said:
42+62=16+36=52. and 52 is not a perfect square. 4 and 6 do not form a counterexample.

a counterexample in the negation of the conjecture.

Correct, and 52 is not a perfect square. 4 and 6 are consistent with the conjecture.

we assume the negation of the conjecture. which is for all natural numbers, a and b, if a and b are both even, then (a^2 +b^2) IS a perfect square.

if we use 4 and 6 as counterexamples we do not get a perfect square, so we have a contradiction...

or are you saying we don't need the negation, just provide 4 and 6 as counterexamples and be finished...?

There is an error.

or maybe we're supposed to prove the conjecture false by counterexample...

using 6 and 8 perhaps.

vinnie said:
or maybe we're supposed to prove the conjecture false by counterexample...

using 6 and 8 perhaps.

Well, sure. It is false, isn't it?

1. What is the definition of a perfect square?

A perfect square is a number that is obtained by multiplying a number by itself. For example, 4 is a perfect square because it is the product of 2 and 2 (2 x 2 = 4).

2. Can you provide an example of two even perfect squares that do not add up to a perfect square?

Yes, for example, 2 and 8 are both even perfect squares (2 x 2 = 4 and 4 x 4 = 16), but their sum (4 + 16 = 20) is not a perfect square.

3. Is it possible for the sum of two even perfect squares to be a perfect square?

Yes, it is possible. For instance, 4 and 16 are both even perfect squares (2 x 2 = 4 and 4 x 4 = 16), and their sum (4 + 16 = 20) is also a perfect square (5 x 5 = 25).

4. What is the mathematical proof that the sum of two even perfect squares is not a perfect square?

The proof is based on the fact that perfect squares always have a square root that is a whole number. If we assume that the sum of two even perfect squares is also a perfect square, then the square root of this sum must also be a whole number. However, since the two even perfect squares have a common factor of 2, their sum will always have a square root that is not a whole number (e.g. the square root of 20 is approximately 4.47). This contradicts our assumption, proving that the sum of two even perfect squares is not a perfect square.

5. Why is it important to prove this statement?

This statement is important because it helps us understand the properties and limitations of perfect squares and their sums. It also has practical applications in fields such as cryptography and number theory.

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