Wild Numbers Report to page 56

[Enoch Haga]

SeqFans,

Neil has asked me to post this message for all interested.

Enoch
__________

Dear Neil,

Schogt's book is actually 152 pages. It begins page 7 and ends page 159. So
far I'm to page 56. I've found enough to indicate trouble with our sequence
which I've extended to a(104). My notation at right is shorthand for a(0)
through a(4), etc.

11 67 2 4769 67 - 0,4
13 78 2 58710 78 - 5,9
15 89 2 69811 89 - 10,14
17 910 2 710912 910 - 15,19
19 1011 2 8111013 1011 - 20,24
21 1112 2 9121114 1112 - 25,29
23 1213 2 10131215 1213 - 30,34
25 1314 2 11141316 1314 - 35,39
27 1415 2 12151417 1415 - 40,44
29 1516 2 13161518 1516 - 45,49
31 1617 2 14171619 1617 - 50,54
33 1718 2 15181720 1718 - 55,59
35 1819 2 16191821 1819 - 60,64
37 1920 2 17201922 1920 - 65,69
39 2021 2 18212023 2021 - 70,74
41 2122 2 19222124 2122 - 75,79
43 2223 2 20232225 2223 - 80,84
45 2324 2 21242326 2324 - 85,89
47 2425 2 22252427 2425 - 90,94
49 2526 2 23262528 2526 - 95,99
51 2627 2 24272629 2627 - 100,104

The first sentence in the book is "Five plus three equals eight." In
general, the text seems a curious mixture of arithmetic and pseudo higher
math. As other reviewers have noted, the book is well-plotted, fast moving,
and  believable enough to bait the reader to the next page.

The characters in the book are some mathematics professors, including Isaac
Swift, who is the one solving the problem, and a psychotic auditing student,
a former high school math teacher.

The Wild Number Problem, posed by Anatole Millechamps de Beauregard, born
1791, had not been solved by the time of his death in 1825 at age 32. On
page 34 of the Schogt's book we have that it "originally wasn't much more
than a tricky arithmetical problem."

Then follows a statement about the method (which unfortunately doesn't
resemble ours): "Beauregard had defined a number of deceptively simple
operations, which, when applied to a whole number, at first resulted in
fractions. But if the same steps were repeated often enough, the eventual
outcome was once again a whole number." This sounds a little like the Ulam
sequence I submitted some time ago -- one of those where you begin with a
number, do some manipulations, and then the number gets back to 1 or the
original number.

According to Beauregard, he believed that a wild number lurks in every
number - p 34 still. This wild number emerges when a number is "provoked"
long enough. Page 34 is where we find the sequence where 0 yields 11, 1
yields 67, 2 yields itself, 3 yields 4769, and 4 yields a repeat of 2.
Therefore a(0)=11 as we have it.

Beauregard it states on page 34 had found 50 wild numbers by the time of his
death, presumably including the first five. But the book doesn't say whether
the numbers were in sequence or not. Prime numbers are said to be especially
difficult to "provoke" (to yield the wild number), and the wild numbers have
a strong proclivity to repeat. It seems that different "initial numbers"
must be applied to different numbers in order to provoke out the wild
numbers. Again this seems like the Ulam sequence mentioned above.

On page 36 we learn that in 1907 Heinrich Riedel proved that 3 could never
be a wild number. This ended speculation that all numbers might be wild.
Unfortunately this also blows my theory that a(2)=2 could at some point
change to a 3. There is no 3 in wild number sequence so it is clasified as a
"tame" number. In 1912 Riedel proved the number of tame numbers to be
infinite. In the 1960's Dimitri Ivanovich Arkanov "showed that there was a
fundamental relationship between wild numbers and prime numbers." Solving
the Wild Number Problem would lead to a breakthrough in number theory.
Dimitri is one of Isaac's professor colleagues.

In 1981, with the aid of a computer, a 30-digit wild number was provoked out
of 103. Thus a 30-digit wild number at a(103) seems to demolish the sequence
that we have constructed (where we have an 8-digit "wild" number).

I'll continue reading thoroughly to the last page and report anything
further I find.

Enoch

P.S. This material reminds me of Euler's table of sums of factors as
described in some book of Polya's that I read a long time ago. Perhaps there
is some connection if this Wild Number Sequence actually is on the esoteric
and exotic side.

Finally, I note that 67^2=4489. Adding the prime factors of 4769, 19+251, we
get 270. 4489+270=4759, which happens to prime and just under 4769. Maybe a
clue, maybe not!