# Prove that the fourth power of an odd integer is expressible in the form 16n + 1 for n ∈ Z.

## Solution to Prove that the fourth power of an odd integer is expressible in the form 16n + 1 for n ∈ Z.

If m is odd,

we can write m = 2k + 1 for some k ∈ Z and so

m^{4} = (2k + 1)^{4} = 16k^{4} + 32k^{3} + 24k^{2} + 8k + 1

and so we have

m = 16(k^{4} + 2k^{3} + (k(3k + 1)/2) + 1.

It remains to show that (k(3k+1))/2 ∈ Z.

If k is even,

then k/2 ∈ Z

while if k is odd,

then 3k + 1 is even and so (k(3k+1))/2 ∈ Z

This completes the proof.