[mindslight]

And yet quantitatively, these "Illegal primes" aren't anywhere near the information content of a movie.

We can try applying your narrative to the number 3.14, supposing laws had made finding the circumference of a circle illegal. Sure, "3.14" isn't illegal on its own (and sometimes it even shows up in the amount of change you're owed). But referencing the number is basically referencing a specific use. And so we can infer that the people broadcasting the number are doing so to highlight how easy finding the circumference of a circle is, and thus induce people to do so.

The informational content of 3.14 is say 12 bits and the original AACS key is 128 bits. Whereas a movie is roughly 10,000,000,000 bits, and even a song is around 32,000,000 bits. These "illegal primes" are very close, if not in, the domain of bona fide facts. And the law generally doesn't criminalize facts (like 3.14), although not for lack of trying!

[JadeNB]

> We can try applying your narrative to the number 3.14, supposing laws had made finding the circumference of a circle illegal.

> The informational content of 3.14 is say 12 bits and the original AACS key is 128 bits.

3.14 is irrelevant for finding the circumference of a circle (from its radius, I guess you meant); it is π that is relevant for finding the circumference, and (conjecturally) the informational content of π is infinite.

[mindslight]

I knew I was bound to get this type of comment. The error resulting from representing pi as 3.14 is much less than the PSNR resulting from representing a feature movie using 10Gb. Be thankful I didn't use "3".

[JadeNB]

> I knew I was going to get this type of comment. The error resulting from representing pi as 3.14 is much less than the PSNR resulting from representing a common movie using 10Gb. Be thankful I didn't use "3".

The comment may have been predictable, but it doesn't seem to me to be invalid. The error resulting from representing π as 3.14 is relatively small numerically, but, in terms of information loss, it is (conjecturally) infinite. (The information encoded in the 1000th decimal place is, information-theoretically though not numerically, just as significant as that encoded in the 1st decimal place.) It therefore seems to me to be false that it is much less than the (finite) error resulting from representing a common movie using 10 Gb.

[mindslight]

Uncompressability doesn't imply information content, for example the output of a stream cipher. I believe the digits of pi can be calculated algorithmically, no? That program is a bound on the information content of pi.

You can make a similar argument about digitizing film via sampling, rather than digitally recording the state of every atom - you're throwing away a huge amount of information. Not infinite of course, but dwarfing what it takes to accurately represent the movie. That's a philosophical curiosity, but not really relevant to the qualitative situation.

My point with 3.14 was that everybody recognizes it as a reference to pi, and it works well enough to calculate the circumference of a circle in an engineering context.

(FWIW I personally hate "Pi Day" as March 14th, as it reeks of numerology. It should probably be the day/time when the Earth has traveled around the sun its distance from the sun)

[JadeNB]

> I believe the digits of pi can be calculated algorithmically, no? That program is a bound on the information content of pi.

Yes, good point. I was using the term informally, and probably incorrectly, to mean that π (conjecturally) contains any finite string of digits (in any base)—what is formally called 'normality' (https://en.wikipedia.org/wiki/Normal_number). Apropos of your correct belief, there's a lovely so called 'spigot algorithm' due to Bailey–Borwein–Plouffe that can generate any hexadecimal digit of π that you please without having to generate intermediate digits: https://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_formula .

[JadeNB]

> FWIW I personally hate "Pi Day" as March 14th, as it reeks of numerology. It should probably be the day/time when the Earth has traveled around the sun its distance from the sun

Incidentally, wouldn't that be the less compelling "1 radian day"? Also, to celebrate such a day, we'd have to decide: travelled since when?, and then we're right back to arbitrariness (why choose January 1, or whatever other date/time is chosen)?

For me, while it's hardly an accurate picture of what math is or why it matters, I'm happy enough to have anything that gives people a positive impression of mathematics.