> (you can't just square amounts of money, just consider how it would work if you changed currency: $8 is 1260¥, squares it makes it 1587000¥ which is $10k != $64)
What can you square then? For example, can you square lengths? E.g. 1km is 1000m, what is its square?
That is because "square length" is its own unit, which we call area. Square money is not meaningful as a unit, that is the problem. You can square anything you want but it turns it into a different unit, which the original commenter did not do (they presumed squaring dollars still gives you dollars back).
1km² aka 1,000,000m², note how the resulting units aren't km and m but square kilometers and square meters.
> What can you square then?
In that case it's the number of operations (which is unitless) that must be squared and then multiplied by the cost of each operation. For instance (figures are completely made up for illustration purpose) if one individual operation costs 0.1 cent, and you have 8000 ops for the factorization, it costs $8, and the operation number squared means you have 64,000,000 operations, and the total cost is $64k. In practice we're talking about trillions of very cheap operations but when you square that number you get an insanely big number which even multiplied by a small individual cost ends up costing trillions of dollars, putting it out of reach of factorization from anyone.
I'll keep trying to explain it to you: you cannot take the square of a price.
If my Yuan exchange rate example didn't convince you, let's have a few thoughts experiments:
- let say you can do can do some amount of work for less than $1 (maybe even factoring a 512 bits number) let's call that amount of work X, and you do it for say $0.9. Do you think you can do X² work for price^2, which is $0.81 ? Yes, much more work for less than the price of doing X, isn't that magical?
- a hard drive with 1TB of storage costs $40. Do you think you can have 1 Yottabyte (10^12 squared is 10^24) of storage for just $1600?
There's a reason to all these paradoxes, it simply makes no sense to take the square of a sum of money, because you can't pay it in square dollars.
> so if a search space of 65536 costs you $8, then a search space of 4294967296 = 65536 x 65536 = 8 x 8 = 8^2 = 64
So close, yet so far: the correct answer here is “65536 x 8 = $525k”, not “8x8 = $64”. If a $8 worth of hard drive can store 65536, then to store 4294967296 you need 65536 of such drives, not 8…
What can you square then? For example, can you square lengths? E.g. 1km is 1000m, what is its square?