In toying around
with my usual mathematics hobby, I discovered a new mathematical constant
some years ago which lies between “e” (the natural logarithmic base;
e = 2.718281828....) and “pi” (the circumference-to-diameter ratio;
pi = 3.141592653....).
My new “constant”,
which I will call “L” is as follows:
L = Summation of (x!/x^x) from x = 0 to infinity.
L = 2.87985386217525853348630614507096....
The number appears
to be “transcendental” (meaning it has no regular or repetitive pattern
of any kind, is irrational, and can not be expressed as a ratio or any
two integers). For those of you familiar with continued fractions,
the first 30 partial quotients are:
[2, 1, 7, 3,
10, 1, 1, 1, 2, 4, 330, 2, 2, 1, 2,
1, 3,
1, 8, 1, 1, 7, 1, 26, 1, 1, 2, 7, 20, 3, ....]
Calculation of the first 10 convergents Of “L”:
[2]
= 2.
= 2
[2, 1]
= 3.
= 3
[2, 1, 7]
= 2.875 = 23/8
[2, 1, 7, 3]
= 2.88
= 72/25
[2, 1, 7, 3, 10]
= 2.87984496.... = 743/258
[2, 1, 7, 3, 10, 1]
= 2.87985865.... = 815/283
[2, 1, 7, 3, 10, 1, 1]
= 2.87985212.... = 1558/541
[2, 1, 7, 3, 10, 1, 1, 1]
= 2.87985436.... = 2373/824
[2, 1, 7, 3, 10, 1, 1, 1, 2] =
2.87985381.... = 6304/2189
[2, 1, 7, 3, 10, 1, 1, 1, 2, 4]
= 2.87985386.... = 27589/9580
(2, 3, 23/8, 72/25, 743/258, 815/283,
1558/541, 2373/824,
6304/2189, 27589/9580)
NOTE: 27589/9580 = 2.8798538622129.... is accurate to the first 10 decimal places.
Does anybody recognize this value as referenced to some aspect of physics, math, or elsewhere? Quite often, “new” math discoveries have been deduced “earlier”, possibly in ancient times by such folk as Archimedes, Pythagoras, and the like, or even any number of modern mathematicians far more famous than myself (meaning virtually any fame at all!). If you find this value familiar in some way, please kindly E-mail me at:
PRIME NUMBERS
Another of my
passions is the study of prime numbers (“primes”). A “prime number”
is defined as: any number divisible only by itself and “1”.
As all the numbers increase in regular order, such primes occur seemingly
at random, with no particular order and no way to predict when and where
they will occur. At this point I might cite the two algebraic equations:
y = x^2 + x + 17 and y = x^2 + x + 41
which have all prime solutions
for “x” between -17 and +17 for the first, and -41 and +41 for the second.
Prime solutions
of the first, starting with x = 0, are:
x y x y
x y
0 17 6 59
12 173
1 19 7 73
13 199
2 23 8 89
14 227
3 29 9 107
15 257
4 37 10 127 16
289* (17 x 17)
5 47 11 149 17
323* (17 x 19)
Prime solutions of the second, starting with x = 0, are:
For those of you who might need it, below is a listing of the first 100 prime numbers in increasing order:x y x y x y
0 41 14 251 28 853
1 43 15 281 29 911
2 47 16 313 30 971
3 53 17 347 31 1033
4 61 18 383 32 1097
5 71 19 421 33 1163
6 83 20 461 34 1231
7 97 21 503 35 1301
8 113 22 547 36 1373
9 131 23 593 37 1447
10 151 24 641 38 1523
11 173 25 691 39 1601
12 197 26 743 40 1681* (41 x 41)
13 223 27 797 41 1763* (41 x 43)
1 29 71
113 173 229 281 349 409
463
2 31 73
127 179 233 283 353 419
467
3 37 79
131 181 239 293 359 421
479
5 41 83
137 191 241 307 367 431
487
7 43 89
139 193 251 311 373 433
491
11 47 97
149 197 257 313 379 439
499
13 53 101 151
199 263 317 383 443 503
17 59 103 157
211 269 331 389 449 509
19 61 107 163
223 271 337 397 457 521
23 67 109 167
227 277 347 401 461 523
One other final
interesting bit on primes. In the simple equation “y = P - 210”,
where “P” defines each ODD prime (excludes “2”, the only EVEN prime), there
is a sequence between P = 89 and 331 that produces a series 42 positive
and negative “y” values that are all primes. What is so unusual about
“210”? It is the product of the first five primes (1 x 2 x 3 x 5
x 7 = 210)!
For those interested,
below is a table of those 42 values for “y = P - 210”.
89 - 210 = -121*
167 - 210 = -43 251 - 210 = 41
97 - 210 = -113
173 - 210 = -37 257 - 210 = 47
101 - 210 = -109
179 - 210 = -31 263 - 210 = 53
103 - 210 = -107
181 - 210 = -29 269 - 210 = 59
107 - 210 = -103
191 - 210 = -19 271 - 210 = 61
109 - 210 = -101
193 - 210 = -17 277 - 210 = 67
113 - 210 = -97
197 - 210 = -13 281 - 210 = 71
127 - 210 = -83
199 - 210 = -11 283 - 210 = 73
131 - 210 = -79
211 - 210 = +1 293 - 210 = 83
137 - 210 = -73
223 - 210 = 13 307 - 210 = 97
139 - 210 = -71
227 - 210 = 17 311 - 210 = 101
149 - 210 = -61
229 - 210 = 19 313 - 210 = 103
151 - 210 = -59
233 - 210 = 23 317 - 210 = 107
157 - 210 = -53
239 - 210 = 29 331 - 210 = 121*
163 - 210 = -47
241 - 210 = 31 337 - 210 = 127
The series of 42 primes is located only BETWEEN the asterisked numbers. The asterisked numbers are, of course, 11^2 (11 x 11) where the series breaks down in both directions. On the other sides of 11^2, the “y” solutions contain both prime and non-prime numbers in haphazard order. Trying this method with higher-sequence products such as 2310 and 30030, products of the first 6 and 7 primes, the results have been uncertain. Perhaps there is a higher-product sequence of primes that will yield a much larger series -- to be discovered by someone else.
This is just the tip of the iceberg with prime numbers. There is much more research to be done -- to try to determine some sort of pattern out of the randomness, or even predict where the next prime number (ahead) will occur.
THE PYTHAGOREAN THEOREM & “TRIPLETS”
Another of my enduring math interests is the Pythagorean Theorem -- in short, expressed “C^2 = A^2 + B^2”. This one theorem alone is the very basis of trigonometry, the study of right triangles. In the movie “The Wizard of Oz”, when Scarecrow first got his “brain” (diploma-mill parchment?) from the guileful Wizard, he made a statement to the effect: “The square of the hypotenuse of an ISOSCELES triangle is equal to the sum of the squares of the other two sides!” (Oz is a great place to visit, but you wouldn't want to live there!)
Now even as a kid, I started wondering about those words. Turns out Scarecrow should’ve said, “....RIGHT triangle....” instead of “isosceles”. There’s only ONE “isosceles” right triangle, and that’s the one where the
other two angles are 45 degrees! Perhaps the screenwriter at the time must’ve thought the word “isosceles” sounded much more “erudite” to (what he must’ve deemed) “naïve” viewers than the very ordinary word “right”. Any other Oz watchers ever catch that?!
As I said, the Pythagorean Theorem is quite simply expressed as:
DIFF = 9
DIFF = 18 DIFF = 25
15 8 17
24 7 25
35 12 37
21 20 29
30 16 34
45 28 33
27
36 45 36
27 45 55 48 73
33 56 65
42 40 58
65 72 97
39 80 89
48 55 73
75 100 125
45 108 117
54 72 90
85 132 157
51 140 149
60 91 109
95 168 193
57 176 185
66 112 130 105 208
233
63 216 225
72 135 153 115 252 277
69 260 269
78 160 178 125
300 325
The most interesting
aspect of all these triplet series is the prominent appearance of many
primes (prime numbers). As such, one of my research projects is to
somehow interrelate the appearances of these primes with their natural
appearances in the integral number sequence. Or, perhaps, there’s
some way to express the appearance of primes via the Pythagorean Theorem.
The research has been compulsive and fascinating; but to date I’ve
found no obvious correlations -- yet. I offer this info to anyone
else who might want to look further into this.
The “DIFF =”
in the headings indicates the fixed (constant) difference value between
the second and third number of each series of triplets. For example,
in the first column (“DIFF = 1”), the third number is ALWAYS one greater
than the second; and in the 5th column (“DIFF = 18”), the third number
is ALWAYS 18 greater than the second.
For those of
you viewing this on a color monitor, all the RED
numbers are primes, and all numbers shown in BLUE
are MULTIPLES of some basic common factor. For example, in the second
column (“DIFF = 2”), with the (6, 8,
10)
triplet: if you divide each number by 2, then you’re back to the
basic (3, 4,
5).
Hence, (6,
8,
10)
is a MULTIPLE of (3, 4, 5).
And in the 4th column (“DIFF = 9”), (63, 216,
225): if you divide each number by 9, then you’re back to the basic (7, 24, 25).
A “primitive” triplet (the majority of those occurring in all the tables) is reduced to its lowest possible factors which have no common divisor. Note also that in the “DIFF = 1” series, ALL the triplets are “primitive”;
while in all the other series, at least SOME of them are MULTIPLES of some other triplets. And for the “DIFF = 3” through “DIFF = 7” and also other “DIFF =” values not shown, for some reason, ALL the triplets are multiples.
I hope at least some of you will find this interesting. And one more last bit on Pythagorean triplets: if the difference between the first and second number were set at “1”, then the solutions become far more “geometric” (i.e., grow RAPIDLY larger!). The short table below shows the first six solutions for the difference of “1” between the FIRST TWO numbers:
3
4 5
20 21 29
119 120 169
696 697 985
4059 4060 5741
23660 23661 33461
GENERAL
I’ve made many
tables, using various setup criteria, showing the distribution and spread
of primes and other numbers in their appearance. If and when I have
the chance, I’ll try to post at least some of those annotated numerical
tables for others to examine and possibly discern more from them than I
could.
One very interesting
“starter” table is the listing for regular multiplication used in many
schools. It is as follows:
1
2 3 4
5 6 7 8 9
10 11 12
2 4
6 8 10 12 14
16 18
20 22 24
3 6
9 12 15 18 21
24 27 30 33 36
4
8 12 16
20 24 28 32 36
40 44 48
5 10 15
20 25 30 35 40
45 50 55 60
6 12 18
24 30 36 42 48
54 60 66 72
7 14 21
28 35 42 49 56
63 70 77 84
8 16
24 32 40 48 56 64
72 80 88 96
9
18 27 36
45 54 63 72 81
90 99 108
10 20 30 40
50 60 70 80 90 100
110 120
11 22 33 44
55 66 77 88 99 110 121
132
12 24 36
48 60 72 84 96 108 120 132 144
The center diagonal
contains all “squares” (multiplying the number by itself): 1,
4,
9,
16,
25,
etc. For those with color monitors, these “squared” numbers are shown
in GREEN. But there are also other “squares”
that occur OUTSIDE that diagonal -- and those also form an interesting
series of placements if this table is continued beyond 12.
Also, there
are also “cubic” numbers (multiplying the number by itself THREE times)
shown in BLUE. The cubic values in the
above table are: 1 (1 x 1 x 1), 8
(2 x 2 x 2), 27 (3 x 3 x 3), and 64
(4 x 4 x 4). The first 10 “cubes” of numbers, in order, are: 1,
8,
27,
64,
125,
216,
343,
512,
729,
and
1000.
And there are also some higher-power numbers scattered through the table: 32 (2^5), 64 (2^6 -- also 4^3 and 8^2), 81 (3^4 -- also 9^2). 4th-power numbers are shown in PURPLE. Note also that some of the numbers can be both “cubes” and “squares”, and other powers of different numbers. (“1”, of course in an “any-power” number because no matter how many times it is multiplied by itself, the answer is always -- “1”! It is shown in GREEN only because of webpage limitations.)
Otherwise, in the table above, where this occurs, such numbers are shown in bi-color. For example, the number 64 is shown in GREEN and BLUE because it is either 8^2 or 4^3. (For those who haven’t figured it out yet, the symbol “^” means “to the POWER of” -- because webpages can’t handle normal EXPONENTS!)
FERMAT’S LAST THEOREM, quite simply stated, is: “For A^n + B^n = C^n, there are no integral (whole-number) solutions for n > 2.” For n = 2, it becomes the Pythagorean Theorem! In other words, there are NO “Fermat Triplets”. Fermat’s Last Theorem supposedly has remained unproven until recently. A few years ago or so there was an article in a scientific magazine about a local scientist/ mathematician who apparently proved its validity via some tedious round-about “algorithm” (don’t ask what that is -- it’s too difficult to explain here!) that took weeks, months, or years on one or more high-power computers.
I read over all the blurb, and while it sounded very convincing, somehow I have the feeling that there’s still a “Fermat Triplet” hiding out there in those larger exponents -- just waiting to be found! And while it took extremely
complicated, tedious methods to “PROVE” its so-called validity, only ONE of those triplets-in-hiding, if and when found, would DISPROVE Fermat’s Last Theorem once and for all. And before the “proof by algorithm” showed up, I did some of my own work on that little dilemma which, hopefully, I might be able to present some day.
ROMAN NUMERALS have always been fascinating to me as a young kid. Whatever I did or didn’t see in them has stuck with me to this day. Many people have posed the question in one form or another: “How do you perform mathematical operations with these seemingly haphazard numbers?” On some old TV science show, the host brought up one of the kids in the audience to his on-set blackboard.
And on it, he wrote something like: “VIII” and “XCV”, and asked the student to add or multiply those two numbers together. The kid looked overwhelmed, probably saying “duhhhhh....” -- with Yours Truly also sharing his impasse. I eagerly awaited the explanation of how to do so, but none came. Instead, there was a brief comment about how IMPOSSIBLE Roman Numerals were to work with in mathematical operations. Were they really? I wondered, and kept thinking about that.
When I finally visited Rome and other parts of the former Roman Empire, I noted all the incredible architectural and other feats the Romans had accomplished with their supposedly unworkable numeral system. And I started thinking harder. And one day recently, it finally came to me -- there WAS a way to add, subtract, multiply, divide, and perform operations with these so-called “unworkable” Roman numerals! By this, I mean “THINKING ROMAN”, which means NOT first converting back to OUR system, doing the math, then converting back to Roman, which I consider outright cheating!
Roman numeral math operations are somewhat more difficult than with our own numbering system, but THEY CAN BE DONE! I showed one of my tutoring students, a young lady who HATED math, multiplication the “Roman” way -- and all she could do was look goggle-eyed at something she’d previously thought IMPOSSIBLE! Her mother, who was also watching, also had an incredible look on her face: “I’ve never seen that done before!”
If and when I get the chance, I’d like to share with you the procedures and rules for ROMAN MATHEMATICS. They’re fairly easy to understand, and I think many of you will find it riveting to learn how the Romans must’ve worked
out their math problems in ancient times. Like I say, it’s a bit harder (though not that much in many cases, for average-sized numbers) than ours (nobody said life was easy in those days!). But once you get into it, it’s FASCINATING! -- a whole new thinking process that wakes us up to “new” old concepts!
In the past, I’ve also tried to make GRAPH MOVIES of equations, attempts of which got lost years ago. An interesting equation “movie” was of “y = x^3 + N(x^2)”. I started at N = -20 and worked up, using increments of
+1, to N = +25. Once “N” is assigned its respective value, the table of “x” and “y” values are calculated, plotted on a graph, a line drawn through to connect the value points, and then the graph is photographed as a single frame.
The movie graph of y = x^3 + N(x^2) shows a very interesting “node” formation in the negative range of “N” which quickly disappears at N = 0. When “N” goes negative, a single dot persists at the x-y axis intersection (0,0), while the main graph is only an odd-shaped vertical curve moving outward in the “-” direction
along the “x” axis. If I can locate the movie attempt I made, I’ll try to post it some day.
I’ve also investigated other forms of curve-fitting, using the aspect of differences (the change from one value to the next) and have come up with some interesting results which I hope to share on the Internet some day.
Solving linear and nonlinear equations with a scientific non-programmable (and also programmable) hand-calculator is something I’ve also figured out. Equations such as “e^x = sin(x) + 3” (where “x” is in radians, in this case) which are difficult to solve by other methods, are easily solved:
“PI” RESEARCHERS: Years ago I was one of those often talked-about researchers of “pi”, the 3.141592653589793238462.... type. Though fascinating, it didn’t produce any worthwhile results. But I’m always still interested!
And I’ve done other odd things like trying to figure out why you can’t trisect an angle in plane geometry. It has to do with the “triple-angle” trigonometric formula having no real solution. And I’ve also looked into the Fibonacci numbers as related to primes, in series analysis, and also to their occurrences in Nature.
****Please E-mail any further questions
to:
“Burt”
<timetravellertwo@netzero.net>
*************************
*************************
Webpage Copyright © 2002 by Burt Libe. Copyright has been registered. Permission is granted to quote from this webpage, provided credit is given to the author/ copyright-holder, along with reference to “TimeTravellerTwo” and this website.
*************************
*************************
CLICK HERE -- to return HOME.
*************************
*************************