6/17/2015
CARNIVAL OF MATHEMATICS, Number 123, Part I
Today is the unveiling of the Carnival of
Mathematics: Number 123. This should be
as "easy as 1, 2, 3"!
Some trivia about 123 that you might find
interesting (or not): (information scoured from the web)
123 is the concatenation of the first three
positive integers. The last two of those integers are prime, and the
first one "used to be prime".
123 is not a prime number. However,
if you start with a 1 and a 3, and put 2, 8, 47, 57, 152, 260, 875, 1178 or
1595 “2”s between the 1 and the 3 you will get a prime number. (1223 is prime,
1222222223 is prime, …)
The STS123 space shuttle mission delivered
the Japanese Kibo Logistics Module and the Canadian Dextre robotics system to
the International Space Station in 2007.
123 is the concatenation of the first three
positive integers. 123 is also the concatenation of the divisors of 23 (1
and 23). 27 when written in base 4 is 123_{4}. 38 when
written in base 5 is 123_{5}. 51 when written in base 6 is 123_{6}.
66 when written in base 7 is 123_{7}. 83 when written in base 8
is 123_{8}. 102 when written in base 9 is 123_{9}.
123 is a 42gonal, alternating, arithmetic,
composite, cyclic, deficient, emirpimes, evil, odd, Lucas, polite, semiprime
and squarefree number.
The totient of 123 is φ = 80. The sum
of its prime factors is 44. The product of its digits is 6, and the sum
of its digits is also 6.
123 is the hypotenuse a unique Pythagorean
triangle: 123^{2} = 120^{2} + 27^{2}. 123 is the
leg of a Pythagorean Triangle example: 123^{2} + 836^{2}
= 845^{2}.
123 is the sum of 3 distinct factorials:
123 = 5! + 2! + 1!. However, 123 cannot be written as the sum of two
squares. 123 can be expressed as the difference of two squares in two
ways: 123 = 22^{2}  19^{2} = 62^{2}  61^{2}.
123 can be expressed as the sum of three squares in two different ways: 123 = 1^{2}
+ 1^{2} + 11^{2}, and 123 = 5^{2} + 7^{2} + 7^{2}.
123 can be expressed as the sum of 4 nonzero squares or as 5 nonzero
squares. 123 can be written as the sum of 6 positive cubes.
It is a Dnumber.
It is a straightline number, since its
digits are in arithmetic progression.
It is a plaindrome in base 7, base 9, base
10, base 14 and base 16. It is a nialpdrome in base 5, base 12, base 13
and base 15.
123 is an equidigital number, since it uses
as many digits as are in its factorization: 123 = 3 * 41.
123 = (1 – 5 + 129), and 123^{2} =
15,129.
123 is the 14^{th} number in the
Toothpick sequence.
123 is the concatenation of the integers
from 1 to 3 (no pun intended).
10^{123} + 3 is a prime number.
123 is the 18^{th} term in a
Fibonacci like sequence that starts with 3, and the next term is the previous
term plus the sum of the digits of the previous term (A016052).
123 is a Sophie Germain semiprime.
2(123) + 1 is also a semiprime.
123 is a semiSophie Germain semiprime (the
product of two Sophie Germain Primes).
123 becomes a prime number if you insert
ones between its digits: 11213.
123 is the 12^{th} term in the
Vtoothpick (or Honeycomb) sequence.
123 is the 17^{th} term in the
Dtoothpick “wide” triangle of the second kind sequences. (A194440)
123 is the 19^{th} term in the
Dtoothpick “narrow” triangle of the second kind sequence.
123 is a pure hailstone number.
There are 123 vertices in Sierpinski’s triangle
after the 4^{th} step.
123 and 119 form a pair of cousin
semiprimes.
123 is a HararyReed number.
123 is a Rhombic number.
123: If you draw a regular 41gon, and
connect each vertex to every other vertex by a straight line segment, 123 of
the regions formed are 9gons (or nonagons).
The divisors of 123 are 1, 3, 41, and
123. The sum of the divisors is 168, and the sum of the proper divisors
is 45.
123 written in Roman Numerals is CXXIII.
123 is a palindromic and undulating number
when written in base 6: 323_{6}.
123 when written in base 7 is 234_{7}.
Both base 10 and base 7 show the
concatenation of three consecutive integers.
123 * 2^{123} – 1 is a prime
number.
The number formed by the concatenation of
odd numbers from 123 down to 1 is prime:
12,312,111,911,711,511,311,110,910,710,510,310,199,979,593,
918,987,858,381,797,775,737,169,676,563,615,957,555,351,494,
745,434,139,373,533,312,927,252,321,191,715,131,197,531.
123 written in Greek is ρκγʹ,
written in Hebrew is קכג, written in Chinese is 一百二十三, and written in Arabic is ١٢٣.
The 123^{rd} prime number is
677. The largest prime smaller than 123 is 113. The smallest prime
larger than 123 is 127.
Omega is 2 (total number of prime
divisors); Tau is 4 (number of divisors); Sigma is 168 (sum of divisors); Phi
is 80 (number of numbers less than 123 that are relatively prime to 123).
The string 123 first occurs in the decimal
expansion of Pi at position 1924, counting from the first digit after the
decimal point. The “3.” is not counted.
123 is the emergency telephone number in
Columbia, the medical emergency telephone number in Egypt, and the electricity
emergency telephone number in Indonesia.
123 is the atomic number of the
yettobediscovered element unbitrium.
123 is 1111011_{2 }when written in
binary.
ASCII code 123 produces a left curly
bracket: {.
The sum of the digits of 123 written in
base 2 (111101_{2}) is equal to the sum of the digits of 123 written in
base 3 (11121_{3}), and also equal to the sum of the digits of 123
written in base 10 (123).
123 is a Belgian0 number, a Belgian2
number, a Belgian3 number, a Belgian6 number, a Belgian8 number, and a
Belgian9 number.
The 123^{rd} pair of Amicable numbers
are 13671735 and 15877065.
The 123^{rd} set (or cycle) of
Sociable numbers are 97438042959521002441065,
108405129946018924066455,
119803210999514749473129, and 108347195621779724136087.
108405129946018924066455,
119803210999514749473129, and 108347195621779724136087.
The 123^{rd} Cake number is 310248
(or 123 straight slices can cut a cake into 310248 pieces).
The 123^{rd} Lazy Caterer number is
7627 (or 123 straight cuts can divide a pizza or pancake into 7627 pieces).
123 can be partitioned in 2552338241 ways.
Mordell's equation (y^{2} = x^{3}
+ n) has no integral solutions when n = 123.
May
3rd is the 123rd day of a nonleap year.
Mathematicians born on this date include: Otto Stolz, Alexander Fraser, Samuel
Volterra, Aryeh Dvoretzky, and Isadore Singer.
During a leap year it is May 2nd.
Mathematicians born on that date include: Étienne Pascal, D'Arcy
Thompson, John Wilton, Kazimierz Zarankiewicz, Gladys Mackenzie, Walter Rudin,
and JacquesLouis Lions.
However,
123 could be written 1/23 (referring to the 1^{st} month, and 23^{rd}
day) or 12/3 (referring to the 12^{th} month, and the third day). Another option to consider is using the
British date format in which 12/3 would refer to the 12^{th} day in the
3^{rd} month. So we also need to
address the dates Jan. 23^{rd}, Mar. 12^{th}, and Dec. 3^{rd}. Mathematicians born on January 23^{rd}
include: John Landen. Ernst Minding, Ernst Abbe. Leopold Klug, David Hilbert, Edwin
P Adams, Anton Suschkevich, and James Lighthill. Mathematicians born on March 12^{th}
include: George Berkeley, Gustav Kirchhoff, Simon Newcomb, Ernesto Cesàro, Lyudmila
Vsevolodovna Keldysh, Gianfranco Cimmino, Vijay Patodi, and Stein Stromme. Mathematicians born on December 3^{rd}
include: Sydney Goldstein, and John Backus.
The smallest Kaprekar number that
starts with 123 is: 1,237,623,712,376,238 =
(153,171,245,343,594 + 1,084,452,467,032,644), and
1,237,623,712,376,238^2 =
1,531,712,453,435,941,084,452,467,032,644.
(153,171,245,343,594 + 1,084,452,467,032,644), and
1,237,623,712,376,238^2 =
1,531,712,453,435,941,084,452,467,032,644.
Below
is a 4 x 4 magic square with a magic constant of 123.
31

28

25

39

26

38

32

27

37

23

30

33

29

34

36

24

All four rows, all four columns, both main diagonals,
all four corners, all four squares in the middle, … well, there are 78 combinations
of 4 squares that add up to 123. I will let
you find them all.
Next is a 9 x 9 Magic Square constructed
with all of the 4 digit integers that can be created using the digits 1, 2, and
3. The number of combinations available
is 81 (3 choices for the first digit, 3 choices for the second digit, 3 choices
for the third digit, and 3 choices for the fourth digit). And it just so happens that a 9 x 9 Magic
Square has to fill 81 boxes. This Magic
Square could also be called an “Unholey” Magic Square since it is constructed
with digits that do not have holes: 1, 2, 3, 5, and 7 (borrowing the definition
of “Holey” primes and “Unholey” primes).
1133

2131

3132

1213

2211

3212

1323

2321

3322

1223

2221

3222

1333

2331

3332

1113

2111

3112

1313

2311

3312

1123

2121

3122

1233

2231

3232

2132

3133

1131

2212

3213

1211

2322

3323

1321

2222

3223

1221

2332

3333

1331

2112

3113

1111

2312

3313

1311

2122

3123

1121

2232

3233

1231

3131

1132

2133

3211

1212

2213

3321

1322

2323

3221

1222

2223

3331

1332

2333

3111

1112

2113

3311

1312

2313

3121

1122

2123

3231

1232

2233

The decimal expansion of the inverse
of 813,008,129,918,699,187 produces a sequence that shows multiples of 123,
written in 10 digit strings, and beginning with 0 * 123.
1/813008129918699187 =
0.
0000000000 0000000123 0000000246 0000000369 0000000492 0000000615 0000000738 0000000861 0000000984 0000001107 0000001230 0000001353 0000001476 0000001599 0000001722 0000001845 0000001968 0000002091 0000002214 0000002337 0000002460 0000002583 0000002706 0000002829 0000002952 0000003075 0000003198 0000003321 0000003444 0000003567 0000003690 0000003813 0000003936 0000004059 0000004182 0000004305 0000004428 0000004551 0000004674 0000004797 0000004920 0000005043 0000005166 0000005289 0000005412 0000005535 0000005658 0000005781 0000005904 0000006027 0000006150 0000006273 0000006396 0000006519 0000006642 0000006765 0000006888 0000007011 0000007134 0000007257 0000007380 0000007503 0000007626 0000007749 0000007872 0000007995 0000008118 0000008241 0000008364 0000008487 0000008610 0000008733 0000008856 0000008979 0000009102 0000009225 0000009348 0000009471 ...
I know that this sequence continues
at least to 47,970 (which is 390 * 123), but that is the limit of my
computing abilities. I suspect that it continues much further. If
you have the needed computing power and can figure out how far this continues
please tell me about it. Thanks, David

The decimal expansion of the inverse
of 999,999,999,999,877 produces a sequence showing the powers of 123,
beginning with 123^0, and written in 15 digit strings.
1/999999999999877 =
0.
000000000000001 000000000000123 000000000015129 000000001860867 000000228886641 000028153056843 003462825991689 …
This sequence does not continue
accurately past what I have shown you above.
The next two terms become so large that they overlap (one of the terms
is over 15 digits long). But we can
fix this problem by adding several 9s to the front of the number listed
above.

(Continued in Part II)
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