It occurred to me, one of the primes ought to be 137, as it is related to the fine structure constant Fine-structure constant - Wikipedia
What other prime would be logically obvious to an alien physicist/chemist/engineer?
The fine structure constant is only approximately 1/137. There is an underlying assumption there than a completely alien species will have a comparable mathematics (and other ways of modeling physical reality) comparable to our own. That is not necessarily the case; an entity that views the world in a fractal manner, or only sees it in statistical distributions may have no real concept of ‘counting numbers’ and Euclidean geometry, and may not recognize our way of representing the fine structure constant (or the Lamb shift, or any other discrete phenomenon). In general, prime numbers are not a good way of trying to establish a fundament of communication of an alien intelligence even assuming that they have the same temporal and spacial sense of the universe that we do.
As an aside, “The Arecibo Message” is my favorite math lesson every year with my fifth graders, where I trick them into thinking deeply about prime numbers and spreadsheets by letting them create their own messages and then decode each others’.
Humans may be best at thinking about finite, concrete objects like the integers, but we’re perfectly capable of thinking about fractals, non-Euclidean geometry, probability distributions, and so on (I assume you got the latter from the Brothers in Anvil of Stars).
Even if some alien race had some native way of thinking that was different, they’d still run across the integers and then the primes. If they understood the universe at all, they’d know that angular momentum is quantized and comes in integer multiples of some value. They’d have to invent the integers if they hadn’t already had them (just as human physicists have done for many esoteric mathematical concepts).
Actually, we don’t know that they would understand integers and primes the the same conceptual way that we do. Quantum mechanics involves quantum states that we characterize by integer or rational fraction values but that is because we use “counting numbers” in our basic arithmetic and conceptualize objects at countable sets of common features, and then try to characterize action at the quantum level in discrete states, which often causes difficulty with supposed ‘paradoxes’ that are really just a consequence of a superposition of probabilities. The reason we conceptualize discrete objects at the macroscopic level is because our visual-spacial sense specifically tries to break up images into patterns of discrete objects and distinct groupings of colors. There is absolutely no expectation than a completely alien form of life will have vision that works in the same way (if they have it at all), and their view of quantum mechanics may be entirely probabilistic.
As far as trying to communicate with an alien species by sending mathematical-based messages, while integers (especially 0 and 1) and the trandenscental numbers (particularly e and pi) are special to us (because our basic counting system is based upon integers and ratios) they may not be at all special to an alien form of life that does not use discrete counting or Euclidian geometry. It would probably be better to use a continuous three dimensional fractal function such as a Rössler system or a set of Apollonian spheres as the fundamental basis to establish artificiality of the signal; the fractal pattern should be discernible whether one interprets the signal in discrete steps or a continuously varying function. How you would encode universally identifiable semantics into such a signal is another question, but then, it is highly unlikely that we would share a sufficiently common basis of experience and perception with an alien species that evolved under different conditions and evolutionary pressures to have any kind of actual ‘conversation’ beyond the most superficial exchange of basic concepts.
Whether they understand the primes the same way we do, a list of them in order is still a consistent pattern that is highly unlikely to come from natural sources.
And it does seem likely that numbers would develop from counting, seeing as we humans are not the only organisms on this planet who count. But, even if they don’t, it seems likely that a sufficiently advanced civilization will eventually hit the integers, even if they get there frmo a different starting place.
There is no way for there to be a perfect message to an unknown entity with an unknown way of thinking. And, if communication is possible at all, we have to have some similarities.
10-11 year olds. I teach the original message and how it works, but the messages they create are a lot simpler–pictures, or words spelled in block letters, and the like. It’s pretty fun.
How much of the message can you as a random human correctly interpret? I know the context of the message, but I’d say at a quick glance I was only able to decipher the meaning of a third of the message. I see the double helix, but I have no idea how they get the constituent elements.
To return to the OP’s question – whether or not an alien intelligence would think along similar lines to us or be able to interpret our “Lite-Brite” message, what would be interesting prime numbers to use?
As Stranger observed, the fine structure constant isn’t exactly 1/137 (although for a long time it was thought to be, and much was made of this), the obvious place to look is mathematics itself.
The first number that occurred to me was 2521. It’s just one more than 2520, which is a number divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, and a lot of other numbers (2520 = 2223335*7). It’s a somewhat large but manageable number that would easily be usable in such mosaic constructions. The number 2519 almost works, but it’s composite, being the product of 11 and 229.
There are a lot of such primes of the form 2 to the n times 3 to the m times… plus r minus one that stand out. Another is 1259 (one less than 2520/2; It turns out that 1261 isn’t prime.)
I’m not clear on what the OP’s means by “one of the primes ought to be 137”. If the implication is that would make the message easier to decode, I don’t see how that’s true. The actual message was 1679 bits, which can only be factored as 73x23 or 23x73. Even if one of the primes were known a priori, it wouldn’t make it significantly easier to factor such a small number.
On the other hand, if the implication is that the receiver would recognize the message as sent from a moderately advanced intelligence because it knows the significance of 137, that also seems dubious because the message is already obviously artificial and sent by an intelligence that has mastered radio communication.
I figured 137, a prime…knowable to any mathematician in the universe…is a value approximately equal to the inverse of a fundamental identity in physics…a value known to Earth scientists for a little over a century now, thus “should” be knowable to an alien radio astronomer scanning their own sky for intelligent signals.
I like the idea of “the other” prime being 2521: part of our message should be establishing the fact we humans by and large use a base ten number system.
A 137x2521 grid would provide a far richer canvas to compose a message, as compared to the coarse pixelated appearance of the Arecibo transmission
The number 2521 in no way implies a base ten number system. It would be prime in any base, and 2520 has the same factors in any base.
If we carry the logic behind using primes to its conclusion, why use two different primes at all? Make your grid the square of a prime number, and there’s only one way to make it a grid.
