COMMENT ⊗   VALID 00011 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.DEVICE XGP
C00003 00003	.COMMENT Heur1: going to extremes
C00004 00004	.COMMENT Chain: plausible scenaria of discoveries
C00005 00005	.COMMENT Chain OVERLAY
C00006 00006	.COMMENT Factorings: exs of divisors
C00007 00007	.COMMENT Complete the Square
C00010 00008	.COMMENT Excerpt: Cardinality
C00013 00009	.COMMENT Excerpt: Cardinality
C00015 00010	.COMMENT Defn of EQUAL
C00016 00011	.COMMENT OVERLAY  Defn of EQUAL
C00018 ENDMK
C⊗;
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.!XGPLFTMAR←100
.MACRO B ⊂ BEGIN NOFILL SELECT 9 INDENT 0 GROUP PREFACE 0 MILLS TURN OFF "{↑↓}[]α" ⊃
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.SELECT 1
.COMMENT Heur1: going to extremes;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Going to Extremes_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓→∞\α" TABS 10,20,50,67,77 SELECT 2


\⊗2A⊗*\\\\⊗2B⊗*




\\\f

\\⊗7a\\b⊗*


\S⊗7↑-↑1⊗*\\\\S



.CENTER


If S is some interesting subset of B,
(e.g.: ↓_extremal_↓), then try to isolate
the elements of A that f maps into S.
.END
.SKIP TO COLUMN 1
.COMMENT Chain: plausible scenaria of discoveries;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Chain of Discoveries_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "→∞\α" TABS 30,50,60


⊗2SETS\NUMBERS→CROSS-PRODUCT⊗*




\\\complete the square



\⊗2MULTIPLICATION⊗*



\\look at the inverse



\⊗2DIVISORS⊗*



\\go to extremes: minimal



\\\⊗2PRIMES⊗*

.END
.SKIP TO COLUMN 1
.COMMENT Chain OVERLAY;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
⊗2.⊗*



.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "→∞\α" TABS 30,50,60,70


⊗2 ⊗*




→inverse



→⊗2PROJEC⊗*







\⊗2 ⊗*



    maximal



 ⊗2MAX-DIVIS⊗*

.END
.SKIP TO COLUMN 1
.COMMENT Factorings: exs of divisors;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Factorings of Numbers_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN OFF "{}" SELECT 2 TABS 7 TURN ON "\"



Factorings-of(7) = 
\{ (7,1) }


Factorings-of(18) = 
\{ (18,1), (9,2), (6,3), (3,3,2) }


Factorings-of(32) = 
\{ (32,1), (16,2), (8,4), (8,2,2), (4,4), (4,2,2), (2,2,2,2) }


Factorings-of(58) = 
\{ (58,1), (29,2) }


.END
.SKIP TO COLUMN 1
.COMMENT Complete the Square;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Complete the Square_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TABS 40,64,75 TURN ON "→∞α{}\" TURN OFF "↑↓"
.SELECT 2

\⊗7count⊗*
Pairs of Sets∞-→α→ Pairs of Numbers
    |\\\|
    |\\\
    |\\\|
    |\\\
    |\\\|
    |\\\
    | ⊗7cross-product⊗*\\\|  ⊗7(?)⊗*
    |\\\
    |\\\|
    |\\\
    |\\\|
    |\\\
    ↓\⊗7count⊗*\\↓
Sets∞-\-∞-\α→ Numbers

.TURN ON "↑↓" SELECT 7


?(x,y) = Count ( Cross-product ( Count↑-↑1(x), Count↑-↑1(y) ) ).


.END
.SKIP TO COLUMN 1
.COMMENT Excerpt: Cardinality;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_As the User Sees It_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0


(1) Looking for examples of things which are EQUAL.
    5 examples were found, and 151 non-examples.
    It is worth trying to generalize the predicate EQUAL.

⊗4USER: Why?⊗*

    Because this heuristic is in the Fillin facet of the Examples concept:
	If the ratio of examples/nonexamples of any predicate P is
	below .1, a generalization of P may be more interesting than P.

(2) Trying to generalize the predicate EQUAL.
  EQUAL has 2 definitions.
  The second definition is recursive.
  It involves the conjunction of 2 recursive calls.
  I can generalize EQUAL by replacing either call by "T".
  The first generalization is EQ1.
  It is like EQUAL, but does not recurse on the CAR of its args.
  The next generalization is EQ2.
  It is like EQUAL, but does not recurse on the CDR of its args.

⊗4USER: Call EQ1 "SAME-LENGTH".⊗*

(3) Looking for examples of things which are SAME-LENGTH.
  25 examples were found, and 100 non-examples.
  It is worth trying to find a canonical form for all objects, an
	operation C(x) which converts them to canonical form,
	so that SAME-LENGTH(x,y) iff EQUAL( C(x), C(y) ).

(4) Trying to canonize objects wrt EQUAL and SAME-LENGTH.
  Conclusion: the canonical args of SAME-LENGTH are 
	BAGS which contain only "T"'s. 
  It is worth restricting Bag-operations to these kinds of Bags.

⊗4USER: Call such Bags "NUMBERS".⊗*

.END
.SKIP TO COLUMN 1
.COMMENT Excerpt: Cardinality;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Maximally Divisible Numbers_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑↓[]{}&"

⊗2N  =  p⊗*↓1↑[a↓1]⊗2p⊗*↓2↑[a↓2]⊗2p⊗*↓3↑[a↓3]...⊗2p⊗*↓k↑[a↓k]

⊗2where   p↓i  is  the  i↑t↑h  prime, 

and   (a↓i + 1) / (a↓j + 1)   =   log(p↓j ) / log(p↓i)

**************************************************

For example:   n could be

2⊗7↑8⊗*3⊗7↑5⊗*5⊗7↑3⊗*7⊗7↑2⊗*11⊗7↑2⊗*13⊗7↑1⊗*17⊗7↑1⊗*19⊗7↑1⊗*23⊗7↑1⊗*29⊗7↑1⊗*31⊗7↑1⊗*37⊗7↑1⊗*41⊗7↑1⊗*43⊗7↑1⊗*47⊗7↑1⊗*53⊗7↑1⊗*
	(which equals 25,608,675,584).

(a↓i + 1)'s  are  (9 6 4 3 3 2 2 2 2 2 2 2 2 2 2 2)

n has 3,981,312 divisors.


AM  Conjecture: 
No  integer < n has that many divisors.

.END
.SKIP TO COLUMN 1

.COMMENT Defn of EQUAL;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Recursive Defn of EQUAL_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓[]{}&" SELECT 2

Definition of ↓_EQUAL_↓(X,Y):

(LAMBDA (X Y)

   (IF  X and Y are both atomic, 
	then X must be identically EQ to Y,

	else

	IF X and Y are both lists,
	then both

	      (CAR(X) is ↓_EQUAL_↓ to CAR(Y))

	      and

	      (CDR(X) is ↓_EQUAL_↓ to CDR(Y))

	]

.END
.SKIP TO COLUMN 1

.COMMENT OVERLAY  Defn of EQUAL;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_._↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑_↓[]{}&" SELECT 2

Definition of ↓_EQ1_↓XX

(

   (

 .
 .	    

 .	
 .

	      XXXXXXXXXXXXXXXXXX  T

	      and

	      (CDR(X) is ↓_EQ1_↓XX to CDR(Y))

	]

.END
.SKIP TO COLUMN 1