Math Newsletter number 25; Wednesday, January 12, 2011.
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What's special about integers 21 through 30?

21 is the smallest number of distinct integer-sided squares
needed to tile a square

21 is the number of spots on a standard cubical die
(1+2+3+4+5+6)

There are 21 letters in the Italian alphabet.
http://www.cyberitalian.com/en/html/alphabet.html

22 is the smallest Hoax number.
http://mathworld.wolfram.com/HoaxNumber.html

23 is the smallest group of people where there is more than a
50% chance that 2 people will share the same birthday (day
and month, not year)
http://en.wikipedia.org/wiki/Birthday_problem

23 is the smallest isolated prime, i.e., not an element of a
set of twin primes. 3,5,7,11,13,17,19 are all elements of
pairs of twin primes.

23 is the smallest prime whose reversal is a power: 32 = 25

23 is the only prime p such that p! is p digits long.

23 is the first prime number in which both digits are prime
numbers and add up to another prime number.

There are 23 letters in the Latin alphabet.
http://www.omniglot.com/writing/latin.htm

23! is the least pandigital factorial, that is it contains
all the digits 0 through 9 at least once
1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5040
8!=40320
9!=362880
10!=3628800
11!=39916800
12!=479001600
13!=6227020800
14!=87178291200
15!=1307674368000
16!=20922789888000
17!=355687428096000
18!=6402373705728000
19!=121645100408832000
20!=2432902008176640000
21!=51090942171709440000
22!=1124000727777607680000
23!=25852016738884976640000

23 Enigma in wikipedia
http://en.wikipedia.org/wiki/23_enigma

There were 23 problems on David Hilbert's famous list of
unsolved mathematical problems, presented to the
International Congress of Mathematicians in Paris in 1900.
http://en.wikipedia.org/wiki/Hilbert%27s_problems

Each parent contributes 23 chromosomes to the start of human
life. The nuclei of cells in human bodies have 46 chromosomes
made out of 23 pairs. Egg and sperm cells in humans have 23
chromosomes which fuse and divide to create an embryo.
http://en.wikipedia.org/wiki/Human_genome

23 is the smallest prime p such that the ring of integers in
the cyclotomic field of pth roots of unity does not have
unique factorization (submitted by Qiaochu Yuan)

23 is the smallest Pillai Prime
(submitted by Jonathan Vos Post)
http://en.wikipedia.org/wiki/Pillai_prime

24 is the largest integer divisible by all positive integers
less than its square root.
4**2 < 24 < 5**2.
24/4 = 6; 24/3=8; 24/2=12; 24/1 = 24

Suppose M>24 were divisible by every positive integer less
than it's square root.
M is divisible by 2.
M is divisible by 3.
M is divisible by 4.
==> M is divisible by 12.
==> M > 35
==> sqrt(M)>5
==> M is divisible by 5
==> M is divisible by 12*5=60
==> M>59
==> sqrt(M)>7
==> M is divisible by 7
==> M is divisible by 60*7=420
==> M >419
==> sqrt(M)>20
==> M is divisible by 11,13,17,19
==> ...
Clearly this pattern repeats indefinitely.

25 is the smallest aspiring number
http://mathworld.wolfram.com/AspiringNumber.html

25 is the smallest square that can be written as a sum of 2
nonzero squares.
http://threesixty360.wordpress.com/2008/02/08/the-one-year-anniversary-carnival-of-mathematics/

26 is the only positive integer to be directly between a
square and a cube.
5**2 + 1 = 26
26 + 1 = 3**3.
It's easy to see that 26 is directly between a square and a
cube.
How can we prove that it is the only integer to be so?
Pierre de Fermat, of famous Fermat's last theorem, proved it.

I will presume that he meant, the square to be less than the
cube. Otherwise, we would have the solution, 0 is directly
between (-1)**3 and 1**2.

y**2 + 2 = x**3

y**2 = x**3 -2

y**2 - 25 = (x**3 - 2) - 25

y**2 - 25 = x**3 - 27

(y-5)(y+5) = (x-3)(x**2 + 3x + 9)

Clearly (x-3) = (y-5) = 0 yields a solution to this
equation.

I do not know how to prove that it is the only solution.

There are 26 sporadic groups in the classification of all
finite simple groups.

The English alphabet has 26 letters.
http://en.wikipedia.org/wiki/English_alphabet

27 is the largest integer that is the sum of the digits of
its cube.

If N is a number such that the sum of the digits of N**3 is
N, the N has remainder 0,1, or 8 when divided by 9.
Proof: 0**3 = 0 mod 9, 1**3 = 1 mod 9, 8**3 = 8 mod 9,
and 2**3 = 8 mod 9, 3**3 = 0 mod 9, 4**3 = 1 mod 9,
5**3 = 8 mod 9, 6**3 = 0 mod 9, 7**3 = 1 mod 9.

10 = 9 + 1

Suppose N were a 3 digit number. Its cube would be at most a
9 digit number. The sum of the digits of a 9 digit number is
less than 100. N is < 100.

N cannot be larger than a 2 digit number.

Suppose N is a two digit number.

We test the possible two digits numbers between 27 and 100.

N = 28, 35,36,37, 44,45,46, 53,54,56, 62,63,64, 71,72,73,
80,81,82, 89,90,91, 98,99

N Sum of digits of N cubed
27 27
28 19
35 26
36 27
37 19
44 26
45 18
46 28
53 35
54 27
55 28
62 26
63 18
64 19
71 26
72 27
73 28
80 8
81 18
82 28
89 35
90 18
91 28
98 26
99 36

28 is the number of dominoes in standard domino sets.

A lunar month is about 29 days.
http://en.wikipedia.org/wiki/Lunar_month

29 is the smallest odd positive integer which is the sum of
squares of three consecutive positive integers.

2**2 + 3**2 + 4**2 = 29

30 is the largest integer with the property that all smaller
integers relatively prime to it are prime.