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.

http://www.facebook.com/pages/Pierre-de-Fermat/118310021523220?v=wall

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.

http://en.wikipedia.org/wiki/Sporadic_group

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.