Show that $ 2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+...+1) $ where b is an odd number and( a, b) are natural numbers

Question

Show that :

$$ 2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+...+1) $$

where b is an odd number and( a, b) are natrual numbers.

So far I’ve tried to use a similar identity as that used to prove the power of a mearsenne prime is a prime but I can’t figure out a way to express this$ -1^k $product infront of the polynomial series. Any hints?

Answer 1

Just expand the right-hand side, and see that almost every term in $$ 2^a\left(2^{ab-a}-2^{ab-2a}+2^{ab-3a}-\cdots+1\right) $$ cancels against a corresponding term from $$ 1\left(2^{ab-a}-2^{ab-2a}+2^{ab-3a}-\cdots+1\right) $$