Easier way of backing up/remembering a bitcoin private key

Question

This is a bit of a general question, knowing already of HD wallets and the mnemonic system for creating, backing-up and using the bitcoin blockchain.

However say you have for example a cold/paper wallet, with just one Bitcoin Private Key. Wouldn't an easier way to back-up/save/remember that address be simply be figuring out where the key lies within the range of possible bitcoin keys and taking note of the exponent of said number? Or Additionally a 10 digit scientific notation number, pretty much the same as remembering a phone number...

Example using python:

from bit import Key

# The exact position of this key is 2^65 in int form: 36893488147419103232
privKey = Key.from_int(36893488147419103232)

print(privKey.address)
print(privKey.segwit_address)

This gives the outputs 1LgpDjsqkxF9cTkz3UYSbdTJuvbZ45PKvx and 3FC7umZDWPvTskbVwG7mn72M8RtW8yFSy7. This is so much easier to remember and technically as long as the key was generated randomly enough wouldn't it be just as secure but easier to remember as opposed to 12/24 words one has to write down somewhere, or the hex format or WIF format private key?

After considering other factors... It wouldn't be all that secure if you want to save a key in the range of all possible keys, some exponents of certain numbers produce floats, I assume eventually one would be limited to specific keys, rendering the security bit a quite useless, but I only tested with exponents of 2 my maths could be wrong.

Say the exponent output was always converted to an exponent of the keys position that gives a non-decimal integer, one could in a sense cover all possible keys in the range no? If so then I stand by my question.

I'm trying to understand why this won't work not proving it works.

Answer 1

Given that private keys need to be random, it's unlikely that a properly chosen private key will be easily represented in the method that you have described. Relying on exponentiation in order to essentially compress the private key will be lossy, or require addition of additional exponentiations. This either leads to reduced security as the private key is no longer uniformly random, or having memorize the entire number anyways, just in a different form.

When it comes to memorization, it's probably easier for people to remember a series of words rather than a sequence of numbers. The words are also able to encode more information per word than per digit in a number, which reduces the number of things actually being memorized. Furthermore, when reading a sequence of numbers, it's easy to lose your spot which can result in memorizing the incorrect number. Lastly, words are also self correcting (with enough letters, the intended word becomes obvious, and BIP 39 is designed such that only the first 3 letters are required), and mnemonics also include a checksum. These make writing down the seed from memory much easier to do.