Just for gigs...

This is a sequel to my earlier post - Numbers: Not everything is magic.

If I show the numbers 1024, 2048, 4096... to a software geek, the immediate answer would be around bits and bytes. But these numbers have another peculiar pattern.

1024 is the first positive integer to satisfy the condition.

aⁿ⁺² + aⁿ = (2a)ⁿ

It is followed by the remaining numbers. Check the following:

1024²² + 1024²⁰ = 2048²⁰

2048²⁴ + 2048²² = 4096²²

4096²⁶ + 4096²⁴ = 8192²⁴

8192²⁸ + 8192²⁶ = 16384²⁶

16384³⁰ + 16384²⁸ = 32768²⁸

……

Now the hack to find it.

aⁿ⁺² + aⁿ = (2a)ⁿ

=>aⁿ (a² + 1) = 2 ⁿ aⁿ

=>a² + 1 = 2 ⁿ

=>a= (2 ⁿ - 1)^1/2

Now, let us find out a way to solve the below equation:

a^n+x + aⁿ = (2a)ⁿ

=>aⁿ (a^x + 1) = 2 ⁿ aⁿ

=>a^x + 1 = 2 ⁿ

=>a= (2 ⁿ - 1)^1/x

So, for n and n+4, the first positive integer to satisfy is 64.

Last but not the least for the post.

a^n+x + aⁿ = (k*a)ⁿ

=>aⁿ (a^x + 1) = k ⁿ aⁿ

=>a^x + 1 = k ⁿ

=>a= (k ⁿ - 1)^1/x

So for k=3 and x = 1, the numbers are 2, 8, 26, 80...