## Wednesday, June 17, 2015

### Carnival of Mathematics Number 123 - Part I

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 STS-123 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 1234.  38 when written in base 5 is 1235.  51 when written in base 6 is 1236.  66 when written in base 7 is 1237.  83 when written in base 8 is 1238.  102 when written in base 9 is 1239.

123 is a 42-gonal, alternating, arithmetic, composite, cyclic, deficient, emirpimes, evil, odd, Lucas, polite, semiprime and square-free 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: 1232 = 1202 + 272.  123 is the leg of a Pythagorean Triangle example:  1232 + 8362 = 8452.

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 = 222 - 192 = 622 - 612.  123 can be expressed as the sum of three squares in two different ways: 123 = 12 + 12 + 112, and 123 = 52 + 72 + 72.  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 D-number.

It is a straight-line 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 1232 = 15,129.

123 is the 14th number in the Toothpick sequence.

123 is the concatenation of the integers from 1 to 3 (no pun intended).

10123 + 3 is a prime number.

123 is the 18th 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 semi-Sophie 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 12th term in the V-toothpick (or Honeycomb) sequence.

123 is the 17th term in the D-toothpick “wide” triangle of the second kind sequences. (A194440)

123 is the 19th term in the D-toothpick “narrow” triangle of the second kind sequence.

123 is a pure hailstone number.

There are 123 vertices in Sierpinski’s triangle after the 4th step.

123 and 119 form a pair of cousin semiprimes.

123 is a Harary-Reed number.

123 is a Rhombic number.

123: If you draw a regular 41-gon, and connect each vertex to every other vertex by a straight line segment, 123 of the regions formed are 9-gons (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: 3236.

123 when written in base 7 is 2347.  Both base 10 and base 7 show the concatenation of three consecutive integers.

123 * 2123 – 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 123rd 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 yet-to-be-discovered element unbitrium.

123 is 11110112 when written in binary.

ASCII code 123 produces a left curly bracket: {.

The sum of the digits of 123 written in base 2 (1111012) is equal to the sum of the digits of 123 written in base 3 (111213), and also equal to the sum of the digits of 123 written in base 10 (123).

123 is a Belgian-0 number, a Belgian-2 number, a Belgian-3 number, a Belgian-6 number, a Belgian-8 number, and a Belgian-9 number.

The 123rd pair of Amicable numbers are 13671735 and 15877065.

The 123rd set (or cycle) of Sociable numbers are 97438042959521002441065,
108405129946018924066455,
119803210999514749473129, and 108347195621779724136087.

The 123rd Cake number is 310248 (or 123 straight slices can cut a cake into 310248 pieces).

The 123rd 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 (y2 = x3 + n) has no integral solutions when n = 123.

May 3rd is the 123rd day of a non-leap 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 Jacques-Louis Lions.

However, 123 could be written 1/23 (referring to the 1st month, and 23rd day) or 12/3 (referring to the 12th month, and the third day).  Another option to consider is using the British date format in which 12/3 would refer to the 12th day in the 3rd month.  So we also need to address the dates Jan. 23rd, Mar. 12th, and Dec. 3rd.  Mathematicians born on January 23rd include: John Landen. Ernst Minding, Ernst Abbe. Leopold Klug, David Hilbert, Edwin P Adams, Anton Suschkevich, and James Lighthill.  Mathematicians born on March 12th include: George Berkeley, Gustav Kirchhoff, Simon Newcomb, Ernesto Cesàro, Lyudmila Vsevolodovna Keldysh, Gianfranco Cimmino, Vijay Patodi, and Stein Stromme.  Mathematicians born on December 3rd 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.

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)

David