<s>[INST] How do I prove that powers of 5 are the sum of two squares using mathematical induction Using mathematical induction I need to prove $$5^n=a_n^2 +b_n^2$$
What I've tried
P(1): $$5^1 = 1^2 +2^2$$
Which is true
P(2): $$5^2 = 3^2 +4^2$$
Which is also true
Now for the induction hypothesis
P(k): $$5^k=a_k^2+b_k^2$$
Then the question suggests to prove for (k+2) which I guess still makes sense.
P(k+2): $$5^{k+2} = 5^2 . 5^k$$
$$=(3^2 +4^2)(a_k^2+b_k^2)$$
This is where I get stuck.
Apologies for formatting I'm fairly new to the site 
 [/INST] Continuing with your induction, assuming that $5^n=a_n^2+b_n^2$ then multiplying both sides by $5^2$ gives $5^{n+2}=(5a_n)^2+(5b_n)^2$. So $5^{n+2}$ is a sum of two squares if $5^{n}$ is. We know it is for $n=2$, so it must be for $n=2+2=4$, etc. Hence by induction, for all even $n \geq 2$ we have that $5^n$ is a sum of two squares. We also know $5^n$ is expressible as the sum of two squares for $n=1$, so it must be for $n=1+2=3$ and by induction for all odd $n \geq 1$. Because a positive integer is either even or odd, we have accounted for all positive integers.
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