Recognizable natural numbers for alien message?

Question

It’s rather cliché that an alien message would use a value like π (actually I expect it to be 2π because Earthlings are weird), but in my story the message which contains the bootstrap information for reading the message has natural numbers as an atomic construct, and fractions, continued fractions, and other ways of expressing a non-whole positive value (let alone a transcendental value!) are built up out of more primitive elements.^※

The lesson on the basic feature of natural numbers can illustrate counting, high enough to show every digit and how positional notation works.^‡

But then it should give some large numbers, so the decoder/receiver can know he’s reading it properly. Now if you were to receive the value of e you’d recognize it as being something special, and out of all the possible numbers you’d know this was “right” because it is distinct and recognizable.

But, what numbers in ℕ are like that? What positive whole number, that’s not too short but needs a bunch of digits, will make the recipient recognize it as being a special somewhat unique value?

If a single value can’t do it, then perhaps a sequence. Primes are too dull and don’t grow fast enough. Something like a list of consecutive powers is too arbitrary.

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Edit: To clarify, this number is not sent/presented in isolation with any need to show artificiality by itself. It appears in a huge message that’s already bootstrapped the main message low-level encoding as “pages” containing images and language encoded as binary files.

This number is to be a “cool” example to conclude the page explaining how natural numbers are encoded in the binary language file.

By “binary language” I mean similarly to how we would store a 32-bit number as 4 bytes (essentially base 256) and how you know to expect a number and how long it is. Not exactly. It’s like digits in some ways… but think of a text file where digits are bytes 0x30 – 0x39, not pictures of what our glyphs look like.

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※ identifiers (“words” and “symbols”) other than numbers, and sentences such as needed to express relationships and algorithms are also build on top of this. Simple expressions like 2+3=5 start on the next page.

‡ details: (spoiler if you’d like to figure it out later when I post the completed image on puzzling, don’t look at the hidden parts.) The message is composed of 6-bit elements (hexets) and the way the channel coding was explained in the simpler/slower/cruder earlier part, it’s clear which code is 000000, 111111 etc. Well, maybe 4-way ambiguity as to the significance of the order and whether the solid mark is a 1 and hollow is 0 or vice-versa.

Anyway, one natural way of assigning numeric values to hexets will show o00 through o56 assigned to “digits” with the same value. A natural number is introduced with hexet o57 and followed by digits in little endian.

The page is ruled into boxes, with many small boxes on one row at the top, getting larger and larger until boxes fill a whole row and then continue to get taller.

Each box has spots in it. They are a few pixels across but vary in size and shape, being irregular in shape. They even have different “colors” (pixel value; reader chose palette arbitrarily). The spots are irregularly positioned as well, clumping here, rarified there.

The first cell has no spots; the next 1, then 2, etc. all the way up to 53 or so,

so it gets into numbers that need 2 hexets to represent. Also in each cell is the label, which is also positioned irregularly in each cell but never touching the rule lines. The labels show o57 o00 in the cell with no spots, o57 o01 in the cell with 1 spot, up through o57 o56 in the cell with 46 spots, then o57 o00 o01 in the cell with 47 spots, through o57 o05 o01 for 52.

You see directly each digit, then that it is positional and in what order.

But it does not carry on counting any higher; large enough to show anything useful would be impractical.

So I think it will go to some sequences after that. The big cool numbers come at the bottom of the page.

Answer 1

I'm interpreting this question as "Is there a large(ish) natural number, the knowledge of which is evidence of advanced maths, and which is in some sense universal"

The theory of groups is of fundamental importance. It arises naturally from the analysis of symmetry, a basic property of nature. Among the groups, some have no normal subgroups, and such a group is said to be simple. Some simple groups are in families, and some are said to be "sporadic".

It is an important fact that there are only a finite number of such sporadic groups, and so there is a largest such group. Given that groups arise naturally, and the size of the largest such group can be calcuated. It provides (in the words of the question) a "positive whole number, that’s not too short" and a mathematically literate species can "recognize it as being a special somewhat unique value". I claim that any sufficiently advanced species would know this value (of course I can't prove that, but the same can be said of any other value: pi, e, or the prime numbers).

The largest such group is called the monster group. It has

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

elements. This sequence of digits, while perhaps not obvious to a non-expert, would be easily recognisable as something special. It shares the properties of pi and e:

• Uniqueness
• Universality
• Distinctiveness
• Importance

Moreover, every digit from 0-9 appears at least once.

There is a sequence of simple sporadic groups, but this attempts to answer the OP question without invoking a sequence, instead with a single large significant number.

So the scene goes:

Scientist A: We thought we had worked out the digits, but then we got this long random sequence. In our notation it is the number 808, ... 000.

Scientist B: I wonder if it means something? [googles] It's the order of the monster group. We are right! We do understand their number system, and we know that they are capable of advanced maths.

(Google only needed if a David Laughlin character is not available)

Answer 2

A sequence of Pythagorean triples should do it.

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17
• 9801, 1980, 9999
• 1001, 501000, 501001

The beginning of the sequence is very recognizable in pattern and works with single digits. The second two are the next two Pythagorean triples, basics of geometry. Then you get into larger ones. The math ($a^2 + b^2 = c^2$) is easy to check and lets you know that you've got the positional notation right. Even better, you can pick arbitrarily large triangles to scale up if there's something special in your notational system at, say, the 1 million mark, or whatever.

Answer 3

I suspect you may be overthinking this. Why not just transmit a simple geometric progression like, say:

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, ...

The first few numbers are small enough that the pattern should be easy to spot, but they increase rapidly. There's a simple one-to-one relation between successive pairs of numbers — the next one is 3 times the previous one — and the numbers in this sequence are also recognizable on their own, being the only numbers not divisible by any prime other than 3.

The base of the progression can indeed be chosen more or less arbitrarily, but I would suggest that it should preferably be:

1. reasonably small, so that the sequence doesn't grow too fast,
2. a prime, so that each number in the sequence has a simple prime factorization, and
3. not equal to (or sharing a common factor with) the base of your number system, to properly exercise the receiver's decoding system.

Thus, for base 2 (or base 2ⁿ) or base 10 numbers, 3 would be a good choice of base for the progression. If you're using e.g. a base 3 number system, the powers of 2 would make a good test sequence. If you're using, say, base 60 like the ancient Babylonians did, try the powers of 7.

Answer 4

First off, I recommend OEIS, the Online Encyclopedia of Integer Seqences. We could post answers all day long here, but OEIS has them catalogued.

Primes may be boring, but don't discount them. They are a very unique feature of the natural numbers. If I was going for clarity, boringness is a virtue. You want it to be boring. Boring and un-misinterpretable.

Sequences are definitely the key. A number without context is meaningless. Sequences naturally give context to every number.

If all you wanted were "big numbers," you could have fun with the Ackermann function. Those get big quick, but they're awfully specialized. Another option might be to have parallel sequences which each give each other context. Consider

x    1   2   3   4   5    ...
x+x  2   4   6   8   10
x*x  1   4   9   16  25
x^x  1   4   27  256 3215

These series grow fast. In addition, you can just keep the pattern going, going through the hyperoperations. You can do tetration (1 4 7625597484987... oh my that jumped quickly!), pentation (A series which grows so fast our positional notation might as well be a primitive counting system which counts "1, 2, many") or any similarly exotic function.

Another twist on primes might be offering prime factorizations of large composite numbers. Given that we rely on that factorization being difficult for RSA encryption, such sets would certainly catch people's interests.

Answer 5

What about the Fibonacci series?

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765...

They come up frequently in nature (at least on Earth), are very easy to calculate and recognize, and grow relatively quickly.

$F_n = F_{n-1} + F_{n-2}$

Answer 6

So, you want a number sequence that someone who has no idea what your numeral system is is able to recognize and extract your numeral system from it? I take it you're using a fixed-base positional system like a normal Type I+ civilization would? Let me suggest ... the factorials.

• Primes grow too slowly for you? Fear not. The factorials will leave the primes, and even the most ambitious of power sequences, in the dust.
  Few sequences grow faster than the factorials, and you want a sequence whose values can be written down anyways, don't you?
• Not too arbitrary. $a_n = \prod_{i=1}^n i$ has no parameters to tweak, except perhaps the $i=1$, and tweaking that doesn't achieve anything good. This increases the chance your recipient makes the right guess and decodes your sequence successfully.
• The factorials have a recognizable pattern in their numeric representation, no matter what numeric base you choose - those long chains of zeroes spanning the half of each number, with new zeroes popping up at regular intervals. Not only does it allow your recipient to recognize the zero, it also allows them to recognize the comma, and even narrow down your base. Very important. Once they know the sequence and the comma, the rest is a breeze.
• There is enough entropy that every digit you have crops up soon enough and can be recognized. You don't want people to be guessing whether 삼 is 3 and 팔 is 8, or vice versa, when they finally crop up in the message itself. The factorials grow so fast you only need about as many numbers as there are symbols in your base. While similar is the case for most sequences and chaos is on your side, some sequences fail spectacularly. 1, 2, 4, 10, 20, 40, 100...? No good.

You might still want to add the sequence of your digits in order to your message. It isn't strictly necessary, but it shows the other side that you're a nice guy - and, if you happen to be using a numeral system with negative-value digits (I'm looking at you, Setunians), this sequence will surely help clear that out.

Answer 7

I was surprised that no one mentioned Catalan Numbers for a possible sequence. I would speculate that the inference of a binary numbering/counting system expressed in a combinatorial sequence suggests an understanding of logic systems and binary computation, but does not assume more advanced knowledge. Though perhaps my view is biased. Admittedly I do not know the history of group theory, though I have seen mention of it briefly when reading about E8 Theory.

Edit

Useful Interpretations of Catalan Numbers

Catalan numbers are presented in OEIS series A000108.

The first few numbers in the series are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, ...

Referring to the number Cn as the nth Catalan number, where { C0, C1, C2, ... } = { 1, 1, 2, ... } here are some interpretations ( quotes are from the wikipedia article ):

• Cn counts the number of expressions containing n pairs of parentheses which are correctly matched:

((())) ()(()) ()()() (())() (()())

• Successive applications of a binary operator can be represented in terms of a full binary tree. (A rooted binary tree is full if every vertex has either two children or no children.) It follows that Cn is the number of full binary trees with n + 1 leaves:

• Cn is the number of different ways a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines (a form of Polygon triangulation). The following hexagons illustrate the case n = 4:

• Interestingly, Spenser Mortensen has described a means of uniquely identifying ( serializing ) any binary tree using Catalan numbers in a sort of hashing function - providing a single numeric value which can uniquely represent the entire structure of a given binary tree.

Rationale for Reducing the Sophistication of the Chosen Symbolic Sequence

1. The maxim, often attributed to Einstein, but to paraphrase, "Everything should be as simple as possible, but no simpler", I think applies here. I would put it this way, let's say it is safe to assume that the recipient of such a message would need some form of computation in order to receive and decode the message, call it a minimum requirement. However, as one sending such a message, I would not assume that the recipient has any more than the necessary knowledge to both receive and decode such a message. If the message itself is the base-line, rather the need to send a message and have it be received is the minimum requirement, then why place any higher requirement on the recipient?

2. Given the nature of the described content that will be transmitted, it seems that a fair portion of it may be amenable to encoding based on the nature of the Catalan Numbers themselves. Certainly anything to do with binary trees or binary operators ( such as representations of recursions of addition, subtraction, multiplication and division ) would be. An perhaps other geometric proofs as well.

Answer 8

The most recognized natural number based off my memories of grade 3 is 58008.

Now, this seems silly, but a serious problem is that we are presuming that the people we are communicating with have similar values and knowledge. It is really, really hard to find a natural natural number; natural numbers in this universe are actually approximations of seemingly continuous phenomena.

As far as we can tell, even the count of protons or neutrons in an atom are just (very accurate) approximation of what "is" is there.

If we limit how alien the aliens are -- they are biological-like organisms built out of chemistry-based life -- then we have a place to start.

Assuming that exponential functions are special to them like they are in our mathematics may not be justified. As a really trivial example, the triangle numbers could be viewed as "as natural" as the square numbers, and have been by previous cultures. The resulting physics may be awkward, but they might be using a different way to model physics than we are. (There are many ways to model human physics; an unknown alien might prefer different ones, or even discover different ones).

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Personally I wouldn't send a large natural number until after I communicated algorithms. Then I'd describe an algorithm that generates a large natural number, and communicate that.

Equations are good, because you can describe how to get the number using math, then repeat the number using a different notation, and do this a few times. Possibly getting the notion of "equals" is key.

Try to communicate using as many strange ways as you can come up with. Issue "rosetta stones" of messages -- many messages that "all mean the same thing". Draw images using various simple formats. Build the foundations of mathematics more than one way. Try to describe things both the reader and writer might have in common, like the emission lines of hydrogen.

Answer 9

Slightly less boring than the primes, faster growing than the factorials, and less obscure than monster groups, are the primorials: Products of the first $n$ primes.

$$p_n\# \equiv \prod_{k=1}^{\infty} p_k$$

Where $p_n$ is the $n$th prime.

The first few primorials, from OEIS: $$ \matrix{ n & p_n\# \\ 0 &\hfill 1 \\ 1 &\hfill 2 \\ 2 &\hfill 6 \\ 3 &\hfill 30 \\ 4 &\hfill 210 \\ 5 &\hfill 2310 \\ 6 &\hfill 30030 \\ 7 &\hfill 510510 \\ 8 &\hfill 9699690 \\ 9 &\hfill 223092870 \\ 10 &\hfill 6469693230 \\ 11 &\hfill 200560490130 \\ 12 &\hfill 7420738134810 \\ 13 &\hfill 304250263527210 \\ 14 &\hfill 13082761331670030 \\ 15 &\hfill 614889782588491410 \\ 16 &\hfill 32589158477190044730 \\ 17 &\hfill 1922760350154212639070 \\ } $$

Note that some of the properties of factorials are missing - the trailing zeros, for example, - but I'm assuming you've got record separators established in an earlier part of the transmission.

One disadvantage is that there isn't any obvious individual number to choose. If it's necessary to use a sequence, this might be a good choice. If the senders don't trust that the receivers have tools like google and OEIS, they could even send a table like this:

$$ \matrix{ n & p_n & \sum p_n & p_n\# \\ 0 &\hfill &\hfill 0 &\hfill 1 \\ 1 &\hfill 2 &\hfill 2 &\hfill 2 \\ 2 &\hfill 3 &\hfill 5 &\hfill 6 \\ 3 &\hfill 5 &\hfill 10 &\hfill 30 \\ 4 &\hfill 7 &\hfill 17 &\hfill 210 \\ 5 &\hfill 11 &\hfill 28 &\hfill 2310 \\ } $$

Where the $\sum p_n$ values are the sums of the first n primes.

(I discovered primorials because they are the cycle lengths of the gaps between rough numbers.)

Answer 10

To show we are able of complex maths, we first of all show that we are able to create a simple message that can be easily understood.

The message I used in this question seems the perfect candidate.

Basically we create a sequence of symbols from which aliens can deduce a sequence of numbers and symbols used for arithmetic operations on those numbers.

This message is just not only a numeric sequence, but it actually shows the alien

• We know what "encoding means"
• We know how to make a encoding meaningfull (they don't have a dictionary to lookup each symbol, so where "symbols starts and ends" it should be obvious)
• We know prime numbers and basic algebra

After that, we agreed on a simple alphabet (or maybe show that we don't want be bound to an alphabet, we could change it everytime, making them aware that we know that alphabet is not important, as long as you can deduce what symbols are used for), we could use it for buildling more complex subjects.

If we just send a plain numeric sequence we are just screaming:

"Hey radio amateur here! Testing, 1, 2, 3"

Instead if we provide something simple but that allows us to build a language, we are clearly showing the intention that we want to communicate and send complex messages.

Also, if the message is simple enough, anyone determined enough could figure its meaning, and even better he could actually learn from it!

Assume actually there's a Remote terminal built by an ancient alien race, then those aliens are died, and there's a new race that actually have no technologic/math knowledge.

However if they see on the terminal something like(our signal sent from far far away):

O   1
OO  2
OOO 3
OOOO 4
O+O  1+1  2
OO+O 2+1  2

They could as well start to learning doing sums and counting, and all based solely on observation.

The interesting fact is that we are not able to send images, speak of how is the weather or what your name is, but we are able to encode math. Axioms, Theorems, Proofs. No matter what their language is, you can (with a really well-though message, starting really simple) encode almost all the math and actually teach it.

Now that I explained the rational, here's the lesson for the aliens. I used our characters to keep that understandable to us, but in reality any symbol could be replaced with another sylmbe without losing the meaningfullness of what I write.

Here is how do we really encodes algebra to arbitrary symbols:

  ____#__1_2_3_4_5_6_7_8_9__0___#_1___##_2___###_3___####_4___#####_5___######_6___#######_7___########_8___#########_9___##########_10___###########_11_##################_18____

The message with "___" replaced by newlines

#__1_2_3_4_5_6_7_8_9__0
#_1
...
##########_10
##################_18

This basically teach to count. There's enough to understand positional notation and to understand what "#" and "0" means.

For the next Lesson we could start doing sums

#__1_2_3_4_5_6_7_8_9__0__+__=

#+#=##
##+#=###
###+###=######
1+1=2
2+1=3
3+3=6
#####+######=###########
5+6=11
9+1=10
99+1=100
23+37=60
5+17=22

Yes, it is very concise with few, but sufficient examples to learn sum. We could also sent a sum table as "extra" of course in our message.

Teaching division/multiplication is the same, I just want to skip that boring part and become functional:

#__1_2_3_4_5_6_7_8_9__0__+__=__x__f__(_)_@

@f(x)=x+1
f(1)=2
f(2)=3
f(4)=5
@f(x)=x+4
f(0)=4
f(1)=5
f(7)=11

Now that we have functions we could define our first axioms

@0
@f(x)=x+1
@0_f(x)
@f(0)=1_1

And after we have our Axioms (almost all Peano Axioms, but induction principle), we have almost defined our natural numbers. Well, in reality it is incomplete, we need first to define "Sets". But without too much effort in being too precise, we did already a lot of setps forward.

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Assuming the aliens use that encoding for their messages, then to be sure someone is reading the message properly I would not use a single number but a expression to check equality, something like:

101*103=10403

Alternatively I would use the page number

1 for page 1

Or a number that represents some function like the number of boxes in the page raised to itself, if there are 10 boxes then then number is 10^10

10000000000

Personally I would choose the last option, because it is not only a number meaningfull in the "page context", but it actually provide usefull as a simple "checksum", once someone reading figure what that number is.

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Answer 11

As Cort Ammon, I'd like to suggest OEIS as the catalog of many important integer sequences the humanity found so far.

More specifically, a good starting place is the core sequences list. From there, I'd pick some sequences which are not based on arbitrary integers (for example, the power series of x is obviously based on x). My list would be something like:

1. Natural numbers or prime numbers. However, you specify you want the numbers to grow faster than that, and maybe a sequence this simple is already covered with the prior setup in the message.

2. Factorials or the Fibonacci sequence. Basic enough to make sense of them even for a person not mathematically inclined.

3. Catalan numbers. They appear naturally in quite a number of ways.

4. Something more abstract, like the number of certain groups or graphs, if you want to show what algebraic structures are familiar to the humanity.