George and the power of two

[PhilW]

Elsewhere on the forum (which I can't link currently as am mobile), George mentioned that:

I also have acquired 4 halves, 2 bottles and 1 magnum of Niepoort.

If indeed George were mildly phobic of any quantity which is not a power of two, then cases of wine must be very tricky. This led to me wondering, what power of three is closest to a power of two, the difference being expressed as a fractional part or percentage of the power of three.

i.e. For what x is (3^x - 2^y)/3^x minimised, for any positive (non-zero, if that needs saying) integer values of x and y?

Corollary: Intuitively, it feels that there should be no limit, with some potential huge values of x and y for which 3^x - 2^y = +/-1, but can this be proven or disproven?

[MigSU]

Interesting topic.
Was about to plot this on a spreadsheet and go up to some very large numbers, just for fun, when I thought "let me ask a couple of LLMs first". This is the answer I got:


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The closest a power of 3 can get to a power of 2 depends on whether you measure closeness by absolute difference (subtraction) or relative ratio (percentage difference).

1. By Absolute Difference (Exactly 1 Apart)
The absolute closest any two distinct integers can be is 1 number apart. There are only three pairs of positive powers of 2 and 3 that achieve a difference of 1:

• (3^1) (3) and (2^1) (2)

• (3^1) (3) and (2^2) (4)

• (3^2) (9) and (2^3) (8)

According to a proven mathematical theorem called Catalan's Conjecture, (3^2) (9) and (2^3) (8) are the only consecutive perfect powers in existence (excluding exponents of 1). There are no higher powers of 2 and 3 that are only 1 apart.


2. By Relative Ratio (Music Theory)
If you are looking at how close large powers get to each other proportionally (which is how the human ear perceives musical intervals), we use the formula (3^n ~~ 2^m):

• (3^5) (243) and (2^8) (256): (3^5) is about 94.9% of (2^8). This forms the basis of the 5-note pentatonic scale.

• (3^12) (531,441) and (2^19) (524,288): (3^12) is 101.36% of (2^19). This exceptionally close overlap (~1.3% error) is why Western music uses a 12-note chromatic scale. The tiny discrepancy between them is known as the Pythagorean comma.

• (3^41) and (2^65): (3^41) is 98.8% of (2^65).

• (3^53) and (2^84): (3^53) is 99.79% of (2^84).

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I found the bolded paragraph really interesting. I would've expected to be one 1 unit apart somewhere along the infinite number line, but apparently not (or at least not according to Catalan's Conjecture).

But yeah, this is what this LLM spit out.