Actually, I don't know it's true or not. I just make observation and write simple python script that checked this for x belongs to [1, 50] and it's alright. Can somebody tell it is true or not and if it is true, how I can prove this?
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You'll have to define the notation - it is not standard. What does the $3^{2^x}-1:2^x$ mean? – Thomas Andrews Feb 19 '24 at 21:32
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I change the notation, sorry for misleading :) – Марат Медведев Feb 19 '24 at 21:36
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3Induction works. Here's a hint:$$3^{2^x} - 1 = (3^{2^{x-1}})^2 - 1 = (3^{2^{x-1}} - 1)(3^{2^{x-1}} + 1).$$ – Theo Bendit Feb 19 '24 at 21:38
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@ThomasAndrews Today I learned of this notation – Benjamin Wang Feb 19 '24 at 21:38