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[ March 1, 2002 ]
Keithing to 2002... an exploration, a search for patterns
Terry Trotter

Prologue

While reflecting on the interesting work being done by many
others in their search for patterns and structures that involve
the current palindromic year 2002, I decided to do the same.
This report will show some of the things that were found by
Patrick and myself.

My method of search might be called "Keithing" on some numbers.
Recall that a Keith Number is defined as follows:

an n-digit integer N with the following property: If a Fibonacci-
like sequence (in which each term in the sequence is the sum of
the n previous terms) is formed, with the first n terms being
the decimal digits of the number N, then N itself occurs
as a term in the sequence. For example, 197 is a Keith
number since it generates the sequence

1, 9, 7, 17, 33, 57, 107, 197, ...

I asked myself: "Why stop at the original number ? Why not continue
the sequence of sums ? Maybe 2002 would occur later." Unfortunately,
it does not appear for 197. Nor, as Patrick's ubasic program showed,
does any 3-digit number produce 2002. (I had earlier verified that no
2-digit number would either.) But things really begin to pick up once
we turned our attention to seed numbers of 4, 5, 6, digits, and so on.

Four-digit numbers

Finally, we hit pay dirt! Patrick's program discovered four
numbers of 4 digits in length. They are given here with their prime
factorization and the sequences of sums that lead up to 2002:

 2622 2 * 3 * 19 * 23 12, 22, 38, 74, 146, 280, 538, 1038, 2002 9 sums 3288 23 * 3 * 137 21, 39, 76, 144, 280, 539, 1039, 2002 8 sums 6647 172 * 23 23, 40, 74, 144, 281, 539, 1038, 2002 8 sums 7295 5 * 1459 23, 39, 76, 143, 281, 539, 1039, 2002 8 sums

Some brief observations can be made at this time
that will have bearing for the next part.

• The numbers 2622 and 7295 are square-free numbers,
i.e. no prime appears more than once.

• When considering only primes of 2 or more digits,
two numbers (2622 and 6647) are multiples of 23.

• The number 3288 is a multiple of 24.

Five-digit numbers

Here we really obtained something worthy of the name of data,
55 five-digit numbers yield 2002. Here they are:

20178, 21961, 22024, 22681, 22753,
22825, 23473, 23545, 23617, 24193,
24265, 24337, 24409, 25057, 25129,
26111, 26840, 26912, 27560, 27632,
27704, 28280, 28352, 28424, 29072,
29144, 29216, 49988, 60185, 60257,
60329, 61049, 62031, 62103, 62760,
62832, 62904, 63480, 63552, 63624,
64272, 64344, 64416, 65064, 65136,
65208, 67711, 68431, 68503, 69151,
69223, 89995, 97987, 98779, 99499.

Now, some statistics...

• Twenty-nine (29) of the numbers are odd, and 26 are even.
They tend to fall in groups, as follows: 1 even, 1 odd, 1 even,
13 odd, 12 even, 6 odd, 12 even, 9 odd.
• All of the 12 evens from 62760 to 65208 are divisible by 24,
whereas none of the other 14 evens is so divisible.
• The nine values that are highlighted in yellow are prime.
• The one highlighted in green (99499) is the lone palindrome,
the only one so far discovered (as of 27/02/02). Though it is
not a prime, its prime factorization -- 29 x 47 x 73 -- is curious,
namely, three 2-digit primes. When they are concatenated to
294773, the result is a prime!
• The only other case where the prime factorization exhibits the
same digital structure of dd x dd x dd is 60329 = 23 x 43 x 61.
(I call that structure ELPF = Equal Length Prime Factors.) Again
the concatenation of those primes as 234361 results in a prime.
• When the prime factorization has this structure - d x dd x ddd -
I call it CLPF for Consecutive Length Prime Factors. The 5 cases
of this are:
• 23545 = 5 x 17 x 277 and 517277 is prime
• 24265 = 5 x 23 x 211 and 523211 = 13 x 167 x 241
• 24409 = 7 x 11 x 317 and 711317 is prime.
• 67711 = 7 x 17 x 569 and 717569 = 739 x 971
• 89995 = 5 x 41 x 439 and 541439 is prime

• There are 8 cases of semiprimes:
22681, 22753, 24193, 25129, 60185, 61049, 68431, 68503.
The 5th one is unique in that its factorization is
5 x 12037, or a 5 with a 5-digit co-prime.
• Two numbers that present an interesting quirk are
27704 and 29144. Both are multiples of 8. Upon dividing
each by 8, we obtain the emirps 3463 and 3643, resp.
• All numbers require from 7 to 9 addition steps to yield 2002.

Summary of data to date [ February 27, 2002 ]

According to Patrick's program, we have these figures:

2-digit numbers : 0
3-digit numbers : 0
4-digit numbers : 4
5-digit numbers : 55
6-digit numbers : 577
7-digit numbers : 7842

The first 4-digit number is 2622 [2 x 3 x 19 x 23]
The first 5-digit number is 20178 [2 x 3^2 x 19 x 59]
The first 6-digit number is 118710 [2 x 3^2 x 5 x 1319]
The first 7-digit number is 1019510 [2 x 5 x 269 x 379]
The first 8-digit number is 10195100 [2^2 x 5^2 x 269 x 379]
The first 9-digit number is 100038299 [1291 x 77489]
The first 10-digit number is 1000380299 [113 x 8852923]
The first 11-digit number is 10003800299 [29 x 344958631]

We might note that even here exist some items of interest:

2622 is a multiple of 19,
20178 is a multiple of 18 and 19,
118710 is a multiple of 18.

The first 8-digit number (10195100) is ten times
the first 7-digit number (1019510).

The final trio of first numbers (ps. all are semiprimes)
show a curious resemblance to each other that their prime
factorization would not suggest at first sight !

100038|299
1000380299
10003800299

Note : starter 38299 is prime !

Epilogue

I began this research in a very elementary manner
with a spreadsheet. While examining cases for 3-digit numbers,
I happened to note that 195 produced 1951, which is the year
in which my lovely wife, Gloria, was born. This coming March 24,
she will celebrate her 51st birthday. Please note that

1951 + 51 = 2002

1951 appears also as substring in
first 7-digit number: 1019510
first 8-digit number: 10195100

Happy Birthday, my dear! From Carlos Rivera I immediately received the following letter:

Just to let you know that the smallest PRIME number of this Keith
type that produces 2002 in ONE STEP (= 1 sum) is the one
I submitted to you before [see year2002.htm].

59999999999999999999999999999999999999999999999999
99999998999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999
If you were just looking for the smallest NUMBER then this is it
49999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999
99999999999999999999999 Terry Trotter and I explored the topic a little further
and found that the following pattern could be extended infinitely.

100038|299
1000380299
10003800299
100038000299

In formula notation we have

10n + 38(0)n-8299  ;  n > 7

Keithing steps needed to reach 2002 is linear n + 7

Steps is (n + 7) and not (n + 8). If we count only the number
of additions as 'steps', it results in a constant 7. The spreading out
of the digits is not a 'step' in the sense of a computational operation.

This must be the most remarkable pattern that the
palindromic year 2002 induced so far !

Factorization of 100038_(n)_299, as n zeros are inserted

100038299 = 1291 x 77489
1000380299 = 113 x 8852923
10003800299 = 29 x 344958631

100038_(3)_299 = 509 x 929 x 211559
100038_(4)_299 = 541 x 4127 x 448057
100038_(5)_299 = 7 x 17 x 47 x 211 x 1117 x 7589
100038_(6)_299 = 2339 x 42769559641
100038_(7)_299 = 110881 x 9022104779
100038_(8)_299 = 431 x 23210672853829
100038_(9)_299 = 61 x 317 x 8963 x 577194929
100038_(10)_299 = 83 x 4547 x 340693 x 7780343
100038_(11)_299 = 7 x 2633 x 542770332591829
100038_(12)_299 = 19 x 419 x 607 x 4181129 x 4951253
100038_(13)_299 = 71 x 9791 x 21262093 x 67682063
100038_(14)_299 = PRIME !
100038_(15)_299 = 6823 x 126781 x 6264173 x 18461701

100038_(16)_299 = 1811 x 552390944229707344009
100038_(17)_299 = 7 x 43 x 587 x 757 x 1093 x 72421 x 944886937
100038_(18)_299 = 109 x 348989 x 2245147 x 1171337325817
100038_(19)_299 = 13802465167 x 72478357155487397
100038_(20)_299 = 347 x 430761942839 x 66926513105303
100038_(21)_299 = 17 x 863 x 839767 x 259611 x 31276819547
100038_(22)_299 = 1511 x 662064857710125744540039709
100038_(23)_299 = 7 x 1429114285714285714285714285757
100038_(24)_299 = 191 x 229 x 112582273 x 20315435574744552817
100038_(25)_299 = 10399 x 23445917970257 x 4103044065166693
100038_(26)_299 = PRIME !
100038_(27)_299 = 233 x 499 x 911 x 10037 x 94099265340888740756371
100038_(28)_299 = 372371 x 1705157 x 5409931 x 291227948713175807
100038_(29)_299 = 7 x 1296343 x 11024198732235879811743394699
100038_(30)_299 =
19 x 29 x 193 x 421x 463 x 17713 x 272458827416457995524807 For n > 2, all values are square free so far. Many primes have been detected :
Using PFGW the following 19 probable primes emerged
— none others below n = 50545 [ PDG, September 29, 2004 ] —
n = 14, 26, 34, 73, 130, 220, 230, 306, 343, 1712, 2954,
7897, 8388, 8856, 9960, 10425, 11341, 13315, 26203
The search will be continued... pfgw -f -lwon128.out won128.txt but look at the case for 23 zeros...
1/7 = .142857...
now look at the colored strings. ( ! )
100038_(23)_299 = 7 x 1429114285714285714285714285757
then note the uncolored digits of the big factor: 14291.......57
1429 - 1 = 1428, then append the 57.
We have the repetend 142857 again ! 7 occurs in every 6th integer as well.
I checked that from the beginning.
Don't forget that 7 is the number of addition sums
needed to yield 2002 in the first place. and the case for 27 zeros... 911.
and all 5 prime factors contain one pair of doubled digits !
100038_(27)_299 =
233 x 499 x 911 x 10037 x 94099265340888740756371 Since Keithing to 2002 leads to the infinite pattern 100038(0)n299
I was wondering if such beautiful patterns would emerge for
other/all values of 'Keithing to N' as well, or is 2002
the odd one out ? Another idea to explore soon...

That reminds me, I still have to search for the first
palprime that Keiths to 2002...  A000128 Prime Curios! Prime Puzzle  Wikipedia 128 Le nombre 128 ```

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