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C00001 00001
C00002 00002 .NSECP(An Example: Discovering Prime Numbers)
C00004 00003 .SSEC(Review of AM)
C00005 00004 . SSSEC(Representation)
C00008 00005 . SSSEC(Agenda and Heuristics)
C00017 00006 .SSEC(What to get out of -- and NOT get out of -- this example)
C00023 00007 .SSEC(Deciphering the Example)
C00029 00008 .SSEC(The Example Itself)
C00044 00009 .GROUP SSEC(Recapping the Example)
C00047 00010 .SSSEC(Math Historian's Recap)
C00053 00011 .SSEC(Delving into the Example)
C00055 ENDMK
C⊗;
.NSECP(An Example: Discovering Prime Numbers)
Here is an excerpt of the output from AM, starting as it investigates
the concept "divisors-of". AM soon defines Primes, and begins to
find interesting conjectures related to them.
.ONCE TURN ON "{}"
After a brief review of AM's control structure in Section {SECNUM}.1,
the reader will be told what the point of this example is -- and is
⊗4not⊗*. Section {SECNUM}.3 provides a few final hints at decoding
the example. The excerpt itself follows in Section {SECNUM}.4. It
is then recapped the way a math historian would report it. Finally, a
few particular pieces of the example are examined in greater detail.
.SSEC(Review of AM)
The reader who is familiar with all the material in Chapter 1
(Overview), or who has actually used AM, may skip this subsection.
. SSSEC(Representation)
AM is a program which expands a knowledge base of mathematical
concepts. Each concept is stored as a particular kind of data
structure, namely as a bunch of properties or "facets" of the
concept. For example, here is a miniature example of a concept$$
"Nos." is an abbreviation for "Numbers". The
vertical bar "|" is a symbol for the predicate "divides evenly into";
the hook "¬" is a symbol for the predicate "the negation of".
"∨" indicates disjunction ("OR"), and the symbol "∀" is read "for all".
Please consult the glossary, Appendix 1b, for fuller discussion of these,
plus other math terms like "Palindromes". $:
.BBOX
~∞ →~
MBOX NAME: Prime Numbers $
~∞ →~
MBOX DEFINITIONS: $
MBOX ORIGIN: Number-of-divisors-of(x) = 2 $
MBOX PREDICATE-CALCULUS: Prime(x) ≡ (∀z)(z|x α→ z=1 ∨ z=x) $
MBOX ITERATIVE: For i from 2 to Sqrt(x), ¬(i|x) $
MBOX $
MBOX EXAMPLES: 2, 3, 5, 7, 11, 13, 17 $
MBOX BOUNDARY: 1, 2, 3 $
MBOX $
MBOX GENERALIZATIONS: Numbers, Nos. with an even no. of divisors, Nos. with a prime no. of divisors $
MBOX $
MBOX SPECIALIZATIONS: Odd Primes, Prime Pairs, Prime Palindromes $
MBOX $
MBOX TIES: Unique factorization, Goldbach's conjecture, Extremes of Number-of-divisors-of $
MBOX $
MBOX INTUITIONS: Primes are the building blocks of all numbers $
MBOX $
MBOX ANALOGIES: $
MBOX CONVERSE: Maximally-divisible numbers are extremes of Number-of-divisors-of $
MBOX INTUITION: Simple groups are the building blocks of all groups $
MBOX $
MBOX INTEREST: Conjectures tieing Primes to TIMES, Divisors-of, or closely related operations $
MBOX $
MBOX WORTH: 800 $
MBOX $
.EBOX
So "creating a new concept" is a well-defined activity, as is filling
in a particular facet of a particular concept.
. SSSEC(Agenda and Heuristics)
An agenda of plausible tasks is maintained by AM. A typical task is
⊗6"Fill-in examples of Primes"⊗*. The agenda may contain hundreds of
entries such as this one. AM repeatedly selects a task from the
agenda and tries to carry it out. This is the whole control
structure! Of course, we must still explain how AM creates plausible
new tasks to place on the agenda,
how AM decides which task will be the best one
to execute next, and how it carries out a task.
Once a task is chosen from the agenda, AM gathers some heuristic
rules which might be relevant to satisfying that task. They are
executed, and then AM picks a new task. While a rule is executing, 3
kinds of "side effects" can occur:
.B04
(i) Facets of some concepts can get filled in (e.g., examples of
primes may actually be found and tacked onto the "Examples" facet of
the "Primes" concept). A typical heuristic rule which might have
this effect is:
.B816
To fill in examples of X, where X is a kind of Y (for some more
general concept Y),
Check the examples of Y; some of them may be examples of X as well.
.END
For the task of filling in examples of Primes, this rule would have
AM notice that Primes is a kind of Number, and therefore look over
all the known examples of Number.
.OO
(ii) New concepts may be created (e.g., the concept Odd-primes may be
deemed worth studying). A typical heuristic rule which might result
in this new concept is:
.B816
If most (but not all) examples of X are also examples of Y (for some
concept Y),
Create a new concept defined as the intersection of those 2 concepts
(X and Y).
.END
When AM notices that most primes are odd, this rule will create a
brand new concept, defined as the set of numbers which are both odd
and prime.
.OO
(iii) New tasks may be added to the agenda (e.g., the current
activity may suggest that the following task is worth considering:
"Generalize the concept of prime numbers"). A typical heuristic rule
which might have this effect is:
.B816
If very few examples of X are found,
Then add the following task to the agenda: "Generalize the concept
X".
.END
Of course, AM contains a precise meaning for the phrase "very few".
When AM looks for primes
among examples of already-known kinds of numbers,
it will find dozens of non-examples for every example of a prime it
uncovers. "Very few" is thus naturally implemented as a
statistical confidence level$$ Philosophers outraged by this should
see section 3.3, wherein are listed objections to this scheme that
even ↓_they_↓ might not perceive at the moment. $.
.END
The concept of an agenda is certainly not new, nor is it limited to
Artificial Intelligence. But one important feature of AM's agenda
scheme ⊗4is⊗* strictly an AI idea: attaching -- and using -- a list
of symbolic reasons to each task which explain why the task is worth
considering, why it's plausible. It is the responsibility of the
heuristic rules to include reasons for any tasks they propose. For
example, let's reconsider the heuristic rule mentioned in (ii) above.
It really looks more like the following:
.B816
If very few examples of X are found,
Then add the following task to the agenda: "Generalize the concept
X", for the following reason: "X's are quite rare; a slightly less
restrictive concept might be more interesting".
.END
.ONCE TURN ON "{}"
If the same task is proposed by several rules, then several different
reasons for it may be present. In addition, one ephemeral reason
also exists: "Focus of attention". Any tasks which are similar to the one
last executed get "Focus of attnetion" as a bonus reason.
AM uses these reasons,
e.g. to decide how to rank the tasks on the agenda. The
"intelligence" AM exhibits is not so much "what it does", but rather
the ⊗4order⊗* in which it arranges its agenda$$ For example,
alternating a randomly-chosen task and the "best" task (the one AM
chose to do) only slows the system down by a factor of 2, yet it
totally destroys its credibility as a rational researcher. This is
one conclusion of experiment 3 (see Section 5.3). $.
AM uses the list of reasons in another way: Once a task has been selected,
the quality of the reasons is
used to decide
how much time and space the task will be permitted
to absorb, before AM quits and moves on to a new task.
This whole mechanism will be detailed in
Section {[2] AGENDASEC}.{[1] AGENDASSEC},
on {"Page" AGENDAPAGE}.
.SSEC(What to get out of -- and NOT get out of -- this example)
The purpose of this example is to convey a bit of AM's flavor. After
reading through it, the reader should be convinced that AM is ⊗4not⊗*
a theorem-prover, nor is it ⊗4randomly⊗* manipulating entries in a
knowledge base, nor is it ⊗4exhaustively⊗* manipulating or searching.
AM is growing a network of data structures representing mathematical
concepts, by repeatedly using heuristics for guidance.
The following points are important but can't be conveyed by any lone
example:
.B04
(i) Although AM appears to have reasonable natural language
abilities, this is a typical AI illusion: most of the phrases AM
types are mere tokens, and the syntax which the user must obey is
unnaturally constrained. For the sake of clarity, I have "touched up" some of the
wording, indentation, syntax, etc. of what AM actually outputs, but left the
spirit of each phrase intact. As the reader becomes more familiar
with AM, future examples can be "unretouched". If he wishes, he may
glance at Appendix 4, which shows some actual listings of AM in
action. In fact, this very excerpt is presented on pages {[3] CH2EX1}
to {[3] CH2EX2}.
.OO
(ii) The reader should be skeptical$$ I shall resist the nasty
temptation to cite examples explaining why such skepticism is
necessary when reading AI theses. $ of the generality of the program;
is the knowledge base "just right" (i.e., finely tuned to elicit this
one chain of behaviors)? The answer is No$$ A provocative note: The
↓_design_↓ of AM was finely tuned so that the answer to this question
would be "No". How "fair" is such a policy? $. The whole point of
this project is to show that a relatively small set of general
heuristics can guide a nontrivial discovery process. Each activity,
each task, was proposed by some heuristic rule (like "look for
extreme cases of X") which was used time and time again, in many
situations. It was not considered fair to insert heuristic guidance
which could only "guide" in a single situation.
.ONCE PREFACE 1
This kind of
generality can't be shown convincingly in one example. Nevertheless,
even within this small excerpt, the same line of development which
leads to decomposing numbers (using TIMES-1-) and thereby discovering
unique factorization, also leads to decomposing numbers (using
ADD-1-) and thereby discovering Goldbach's conjecture.
The same heuristic which caused AM to expect that unique factorization
will be useful, also caused AM to suspect that Goldbach's conjecture will
be useless.
.END
Let me reemphasize that the "point" of this example is ⊗4not⊗* the
specific mathematical concepts, nor the particular chains of plausible
reasoning AM produces, nor the few flashy conjectures AM spouts, but rather an
illustration of the ⊗4kinds⊗* of things AM does.
We shall return to this example later in this
chapter, and again in Chapter 5, to
explain
⊗4how⊗* and ⊗4why⊗* AM did each step. Snapshots will provide glimpses
of the state of knowledge of AM at various moments.
Several more examples will be analyzed in Chapter 5.
A comprehensive discussion of the kind of things which AM can and
cannot do is found in Section 6.2.
.SSEC(Deciphering the Example)
Recall that in general, each task on the agenda will have several
reasons attached to it. In the example excerpt, the reasons for each
task are printed just after the task is chosen, and before it's
executed.
AM numbers its activities. Each time a new task is chosen, a counter
is incremented. The first task in the example excerpt is labelled
*.*TASK 65., meaning that the example skips the first 64 tasks which
AM selects and carries out.
In the example itself, several irrelevant tasks have been excised$$
This is fair, despite the results of Experiment 3, Section 5.3,
because the remaining tasks clump together in twos, threes, etc; they
are uninterrupted lines of research (e.g., Tasks 65-67), separated by
large gaps (e.g, the jump from Task 67 to 79). $. About half of
those omitted tasks were interesting in themselves, but all of them
were tangential or unrelated to the development shown. The reader
can tell by the global task numbering how many were skipped. For
example, notice that the excerpt jumps from TASK 67 to TASK 79.
To help gauge AM's abilities, the reader may be interested to know
that AM defined "Numbers" during Task 26, and "TIMES" was defined
during Task 39. AM started with no knowledge of numbers, and only
scanty knowlege of sets and set-operations. TASK 2, e.g., was to fill
in examples of Sets.
The concepts that AM talks about are self-explanatory -- by and
large. Below are discussed some nonstandard ones.
⊗4↓_BAG_↓⊗* is another word for multiset. A bag is a list-structure
which may contain repeated elements, but which is nevertheless
unordered. A bag is denoted by enclosure within parentheses, just as
sets are within braces. So the bag containing X and four Y's might be
written (X Y Y Y Y), and would be considered indistinguishable from
the bag (Y Y X Y).
↓_⊗4TIMES⊗*-1-_↓ is a particular relation. For any number x,
TIMES-1-(x) is a set of bags. Each bag contains some numbers which,
when multiplied together, equal x. For example, TIMES-1-(18) = {
(18) (2 9) (2 3 3) (3 6) }. Checking, we see that multiplying, e.g.,
the numbers in the bag (2 3 3) together, we do get 2x2x3=18.
TIMES-1-(x) contains all possible such bags (containing numbers >1).
⊗4↓_ADD-1-_↓⊗1 is a relation analogous to TIMES-1-. For any number x,
ADD-1-(x) is also a set of bags. Each bag contains a bunch of
numbers which, when added together, equal x. For example, ADD-1-(4)
= { (4) (1 1 1 1) (1 1 2) (1 3) (2 2) }. ADD-1-(x) contains all
possible such bags (containing numbers >0).
⊗4↓_Divisors-of_↓⊗* is a more standard relation. For any number x,
Divisors-of(x) is the set of all positive numbers which divide evenly
into x. For example, Divisors-of(18) = {1 2 3 6 9 18}.
The definitions for most of the mathematical terms used in the
excerpt can be found in the Glossary (Appendix 1a). Whenever there is
a conflict between "computer science jargon" and "math jargon", I
have opted for the latter. So, e.g., all "functions" are necessarily
single-valued for each member of their domain.
AM is an ⊗4interactive⊗* computer program. It prints out phrases
indicating what it is doing at each moment, and a human being,
referred to as the ⊗4User⊗*, watches AM's activities. At any moment,
the user can interrupt AM to ask questions or re-direct its energies.
In the excerpt, the user never re-directs AM, although he does
occasionally ask AM to rename some concept and to explain its
behavior (e.g., just after *.* TASK 106.). For the benefit of the
reader, I have italicised all the user's comments.
.SSEC(The Example Itself)
.BEGIN NOFILL PREFACE 0 TURN OFF "{}" TURN ON "↑↓\" TABS 18,21 SELECT 3
***Task 65.
Filling in examples of the following concept: "Divisors-of".
3 Reasons:\(1) No known examples for Divisors-of so far.
\(2) TIMES, which is related to Divisors-of, has recently become very interesting.
\(3) Focus of attention: AM recently defined Divisors-of.
26 examples found, in 9.2 seconds. e.g.: Divisors-of(6)={1 2 3 6}.
***Task 66.
Considering numbers which have very small sets of Divisors-of (i.e., very few divisors).
2 Reasons:\(1) Worthwhile to look for extreme cases.
\(2) Focus of attention: AM recently worked on Divisors-of.
Filling in examples of numbers with 0 divisors.
0 examples found, in 4.0 seconds.
Conjecture: no numbers have 0 divisors.
Filling in examples of numbers with 1 divisors.
1 examples found, in 4.0 seconds. e.g.: Divisors-of(1)={1}.
Conjecture: 1 is the only number with 1 divisor.
Filling in examples of numbers with 2 divisors.
24 examples found, in 4.0 seconds. e.g.: Divisors-of(13)={1 13}.
No obvious conjecture. This kind of number merits more study.
Creating a new concept: "Numbers-with-2-divisors".
Filling in examples of numbers with 3 divisors.
11 examples found, in 4.0 seconds. e.g.: Divisors-of(49)={1 7 49}.
All numbers with 3 divisors are also Squares. This kind of number merits more study.
Creating a new concept: "Numbers-with-3-divisors".
***Task 67.
Considering the square-roots of Numbers-with-3-divisors.
2 Reasons:\(1) Numbers-with-3-divisors are unexpectedly also perfect Squares.
\(2) Focus of attention: AM recently worked on Numbers-with-3-divisors.
All square-roots of Numbers-with-3-divisors seem to be Numbers-with-2-divisors.
e.g., Divisors-of( Square-root(169) ) = Divisors-of(13) = {1 13}.
Formulating the converse to this statement. Empirically, it seems to be true.
The square of each Number-with-2-divisors seems to be a Number-with-3-divisors.
This is very unusual. It is not plausibly a coincidence. (Chance of coincidence is below .001)
Boosting interestingness factor of the concepts involved:
Interestingness factor of "Divisors-of" raised from 300 to 400.
Interestingness factor of "Numbers-with-2-divisors" raised from 100 to 600.
Interestingness factor of "Numbers-with-3-divisors" raised from 200 to 600.
⊗4USER: Call the set of numbers with 2 divisors "Primes".⊗*
***Task 79.
Examining TIMES-1-(x), looking for patterns involving its values.
2 Reasons:\(1) TIMES-1- is related to the newly-interesting concept "Divisors-of".
\(2) Many examples of TIMES-1- are known, to induce from.
Looking specifically at TIMES-1-(12), which is { (12) (2 6) (2 2 3) (3 4) }.
13 conjectures proposed, after 2.0 seconds. Testing them on other examples of TIMES-1-.
Only 2 of these 13 conjectures are verified for all examples of TIMES-1-:
Conjecture 1: TIMES-1-(x) always contains a singleton bag.
e.g., TIMES-1-(12), which is { (12) (2 6) (2 2 3) (3 4) }, contains (12).
e.g., TIMES-1-(13), which is { (13) }, contains (13).
Creating a new concept, "Single-times".
Single-times is a relation from Numbers to Bags-of-numbers.
Single-times(x) is all bags in TIMES-1-(x) which are singletons.
e.g, Single-times(12)={ (12) }.
e.g, Single-times(13)={ (13) }.
Conjecture 2: TIMES-1-(x) always contains a bag containing only primes.
e.g., TIMES-1-(12), which is { (12) (2 6) (2 2 3) (3 4) }, contains (2 2 3).
e.g., TIMES-1-(13), which is { (13) }, contains (13).
Creating a new concept, "Prime-times".
Prime-times is a relation from Numbers to Bags-of-numbers.
Prime-times(x) is all bags in TIMES-1-(x) which contain only primes.
e.g, Prime-times(12)={ (2 3 3) }.
e.g, Prime-times(13)={ (13) }.
***Task 80.
Considering the concept "Prime-times".
2 Reasons:\(1) A conjecture about Prime-times will tell us more about Primes and TIMES-1-.
\(2) Focus of attention: AM recently defined Prime-times.
Looking specifically at Prime-times(48), which is { (2 2 2 2 3) }.
4 conjectures proposed, after .6 seconds. Testing them on other examples of Prime-times.
Only 1 of these 4 conjectures are verified for all examples of Prime-times:
Conjecture 1: Prime-times(x) is always a singleton set.
That is, Prime-times is a function, not just a relation.
e.g., Prime-times(48), which is { (2 2 2 2 3) }, is a singleton set.
e.g., Prime-times(47), which is { (47) }, is a singleton set.
This holds for all 17 known examples of Prime-times. (Chance of coincidence is .0001)
This fails for 2 of the boundary cases (extreme numbers): 0 and 1.
Conjecture is amended: Each number >1 is the product of a unique bag of primes.
I suspect that this conjecture may be very useful.
⊗4USER: Call this conjecture "Unique factorization conjecture".⊗*
***Task 84.
Examining ADD-1-(x), looking for patterns involving its values.
2 Reasons:\(1) ADD-1- is analogous to the newly-interesting concept "TIMES-1-".
\(2) Many examples of ADD-1- are known, to induce from.
Looking specifically at ADD-1-(6), which is { (1 1 1 1 1 1) (1 1 1 1 2) (1 1 1 3) (1 1 2 2)
(1 1 4) (1 2 3) (1 5) (2 2 2) (2 4) (3 3) }.
17 conjectures proposed, after 3.9 seconds. Testing them on other examples of ADD-1-.
Only 11 of these 17 conjectures are verified for all examples of ADD-1-:
Conjecture 1: ADD-1-(x) never contains a singleton bag.
Conjecture 2: ADD-1-(x) always contains a bag of size 2 (also called a "pair" or a "doubleton").
e.g., ADD-1-(6) contains (1 5), (2 4), and (3 3).
e.g., ADD-1-(4) contains (1 3), and (2 2).
Creating a new concept, "Pair-add".
Pair-add is a relation from Numbers to Pairs-of-numbers.
Pair-add(x) is all bags in ADD-1-(x) which are doubletons (i.e., of size 2).
e.g, Pair-add(12)={ (1 11) (2 10) (3 9) (4 8) (5 7) (6 6) }.
e.g, Pair-add(4)={ (1 3) (2 2) }.
Conjecture 3: ADD-1-(x) always contains a bag containing only 1's.
.
.
.
Conjecture 10: ADD-1-(x) usually (but not always) contains a pair of primes.
e.g., ADD-1-(10) contains (3 7), and (5 5).
e.g., ADD-1-(4) contains (2 2).
Creating a new concept, "Prime-add".
Prime-add is a relation from Numbers to Pairs-of-numbers.
Prime-add(x) is all bags in ADD-1-(x) which are pairs of primes.
e.g, Prime-add(12)={ (5 7) }.
e.g, Prime-add(10)={ (3 7) (5 5) }.
e.g., Prime-add(11) = { }
.
.
.
***Task 106.
Considering the set of numbers for which Prime-add is defined (has non-empty value).
1 Reason:\(1) Prime-add often has non-empty value. Worth isolating that case.
Warning: no task on the agenda has an interestingness value above 200!!!
Creating a new concept "Prime-addable".
Prime-addable is a kind of Number. x is Prime-addable if Prime-add(x) is non-empty.
Will spend 5.0 seconds filling in examples of Prime-addable.
18 examples found. Here are some of them: 4 5 6 7 8 9 10 12 13 14 17 16 18.
Empirically, all even numbers are also in this set (ignoring boundary cases: 0 2 4 6).
So conjectured. Danger: must examine boundary cases: the numbers 0, 2, 4, and 6.
Two exceptions noticed. The only exceptions are the smallest boundary cases: 0, 2.
Conjecture is amended: All even numbers >2 are the sum of two primes.
Warning: I expect this conjecture will be cute but useless.
⊗4USER: Why?⊗*
.BEGIN FILL PREFACE 0 SINGLE SPACE INDENT 0,2,0 COMPACT
Because the concept of "Primes" originally arose as extreme cases of Divisors-of.
The more closely an operation X is related to the concept Divisors-of, the more
natural will be any conjecture involving both that operation X and Primes.
E.g., conjectures involving both Primes and Times will be natural and useful.
But this conjecture, which involves Primes and ADD-1-, will be cute but useless,
since the relation "ADD-1-" is unrelated to the relation "Divisors-of".
End of explanation.
.END
⊗4USER: Call this conjecture "Goldbach's conjecture".⊗*
***Task 107.
Considering the set of numbers for which the relation Prime-add is single-valued.
3 Reasons:\(1) Prime-add often has singleton value. Worth isolating that case.
\(2) Restricted to this set, Prime-add would be a function.
\(3) Focus of attention: AM recently worked on Prime-add.
Creating a new concept "Uniquely-prime-addable".
"Uniquely-prime-addable" is a kind of Number. x is Uniquely-prime-addable if Prime-add(x) is a singleton.
Will spend 10.0 seconds filling in examples of Uniquely-prime-addable.
11 examples found. Here are some of them: 4 5 7 8 9 12 13.
No obvious conjecture derived empirically.
Will forget "Uniquely-prime-addable numbers", if no Ties found in near future.
.END
.GROUP SSEC(Recapping the Example)
.SSSEC(Schematic Diagram of the Development)
.BEGIN SELECT 8 NOFILL PREFACE 0 MILLS TURN OFF "α↑↓"
Divisors-of
/
/
/ ⊗4Look for extremals⊗*
(very few divisors)
/
/
/
Primes
/\
/ \
/ \
/ \
/ \
⊗4Found some connections⊗*
(to ADD-1-) (to TIMES-1-)
/ \
/ \
Goldbach⊗1'⊗*s Unique
Conjecture Factorization
.END APART
.ONCE PREFACE 2
This little picture symbolizes the reasoning behind the three major
concepts derived in the example. The italicized phrases labelling
the arcs represent the key heuristics used in the plausible
reasoning; the nodes represent the concepts (or the relationships)
discovered and worked on.
The reader may notice that a symmetry is missing: what about numbers
with "very many" divisors, instead of "very few"? AM doggedly thought
that such numbers might be very interesting, and eventually hinted at
what is probably the first explicit characterization of such
numbers$$ See Section {[2] MAXDIVSEC}.{[1] MAXDIVSSEC}, on page {[3]
MAXDIVPAGE}. These are the so-called "highly-composite" numbers of
Ramanujan. They are the local maxima of the function d(n), the
number of divisors of n. Prior to "AM's conjecture", most work on
d(n) was not in characterizing its maxima, but rather, e.g., in
finding its max order, normal order, etc. $.
.SSSEC(Math Historian's Recap)
Let's once again eavesdrop on a mathematician, as he describes to a
colleague what AM did.
This example was preceded by the momentous discoveries of
multiplication and division. Several interesting properites of these
operations were noticed. The first task illustrated (*.*Task 65.)
involves exploring the concept of "divisors of a number" (meaning all
positive integers which divide evenly into the given number). After
tiring of finding examples of this relation, AM investigates extreme
cases: that is, it wonders which numbers have very few or very many
divisors.
AM thus discovers Primes in a curious way. Numbers with 0 or 1
divisor are essentially nonexistent, so they're not found to be
interesting. AM notices that numbers with 3 divisors always seem to
be squares of numbers with 2 divisors (primes). This raises the
interestingness of several concepts, including primes. Soon (*.*TASK
79.), another conjecture involving primes is noticed: Many numbers
seem to factor into primes. This causes a new relation to be defined,
which associates to a number x, all prime factorizations of x. The
first question AM asks about this relation is "is it a function?".
This question is the full statement of the unique factorization
conjecture: the fundamental theorem of arithmetic. AM recognized the
value of this relationship, and was quite excited about it.
In a similar manner, though with lower hopes, it noticed some more
relationships involving primes, including Goldbach's conjecture. AM
quite correctly predicted that this would turn out to be cute but of
no future use mathematically.
The last activity mentioned (*.*TASK 107.) shows AM examining a
rather nonstandard concept: "numbers which can be written as the sum
of a pair of primes, in only one way". These are termed
"uniquely-prime-addable" numbers. It was mildly unfortunate that AM
gave up on this concept before noticing that p+2 is
uniquely-prime-addable, for any prime number p, and that in fact
these are the only even uniquely-prime-addable numbers. The session
was repeated once, with a human user telling AM explicitly to
continue studying this concept. AM did in fact construct
"Uniquely-prime-addable-even-numbers", and then notice this
relationship. Here we see an example of unstable equilibrium: if
pushed slightly this way, AM will get very interested and spend a lot
of time working on this kind of number. Since it doesn't have all the
sophistication (i.e., compiled hindsight) that we have, it can't know
whether what it's doing will be fruitless.
.ONCE TURN ON "{}"
Although not shown by the example, AM undertook the analogous
investigation of numbers with ⊗4very many⊗* divisors. It is worth
noting that most number theorists have ignored studying this variety
of number, with one shining exception: Ramanujan, the self-taught
Indian genius. Both he and AM were mathematically naive enough to
study these numbers out of "symmetry" considerations, since they are
the conceptual converse of prime numbers (numbers with many, not few,
divisors). The story of this adventure must wait until
Section {[2] MAXDIVSEC}.{[1] MAXDIVSSEC}, on page {[3] MAXDIVPAGE}.
.SKIP 5
.SSEC(Delving into the Example)
Before moving on to describe the design of AM, let's squeeze all we can
out of this example.
<< Delve into this example a little. >
< More detailed description of parts of the example, motivating the
design details in the next chapter. >
< Summarize the points to remember about this example. >
< Review the major conclusions just drawn, which will be explicitly
discussed in the next chapter. >