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

m⁴ = (2k + 1)⁴ = 16k⁴ + 32k³ + 24k² + 8k + 1

and so we have

m = 16(k⁴ + 2k³ + (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.