Impossible to have solution to the equation. Induction Proof Problem

[M1ZeN]

Homework Statement

For all integers n, it is impossible to have a solution to the equation

4^n = a^2 + b^2 + c^2

where a, b and c are all positive integers. (Hint: Notice that 4^n = 2^2n is a perfect square. Show (prove) that if m^2 = a^2 + b^2 + c^2, then we must have that a, b and c are all even. This can be done without induction; just think about what remainders a perfect square can leave when divided by four.)

Homework Equations

None in accordance from the chapter. I'm in a discrete mathematics course and this problem's chapter is on Induction. With the previous chapter before it was dealing with standard logic proofs.

The Attempt at a Solution

The only idea I could think of to attempt with is using the proof method of induction just like how I have used in the previous problems of my homework. By induction proof method, I mean the procedure of plugging in "n=1", to see if the proposition is true and continuing after by plugging in "n+1" etc. I'm just not sure how to adjust this problem to solve it by using induction.

 

[Dick]

Homework Statement

For all integers n, it is impossible to have a solution to the equation

4^n = a^2 + b^2 + c^2

where a, b and c are all positive integers. (Hint: Notice that 4^n = 2^2n is a perfect square. Show (prove) that if m^2 = a^2 + b^2 + c^2, then we must have that a, b and c are all even. This can be done without induction; just think about what remainders a perfect square can leave when divided by four.)

Homework Equations

None in accordance from the chapter. I'm in a discrete mathematics course and this problem's chapter is on Induction. With the previous chapter before it was dealing with standard logic proofs.

The Attempt at a Solution

The only idea I could think of to attempt with is using the proof method of induction just like how I have used in the previous problems of my homework. By induction proof method, I mean the procedure of plugging in "n=1", to see if the proposition is true and continuing after by plugging in "n+1" etc. I'm just not sure how to adjust this problem to solve it by using induction.

The way you want to solve it is not quite the same as that form of induction. If n=0 then there is no solution. It's easy to check for n=1 there is also no solution. So if there is one, there must be a SMALLEST value of n that gives a solution. If you've figured out why a, b and c must all be even then you should be able to show that there is also a solution for a SMALLER value of n. That would be a logical contradiction of n being the SMALLEST. Hence there is no solution. Can you fill in the details?