## Thursday, 28 February 2013

### THURSDAY, 28 FEBRUARY 2013

Today is the $59^{th}$ day of the year.

$59$ is prime.

$59 = 1^2 + 3^2 + 7^2 = 3^2 + 5^2 + 5^2$

$(59, 1740, 1741)$ is a Pythagorean triple.

$59$ is a vile number, see A003159 and AviezriS.Fraenkel's original paper.

$59$ is a safe prime because $\frac {59 - 1}{2} = \frac {58}{2} = 29$ which is prime. $29$ is a Sophie Germain prime, see here and here.

## Wednesday, 27 February 2013

### WEDNESDAY, 27 FEBRUARY 2013

Today is the $58^{th}$ day of the year.

$58 = 2 \times 29$

$58 = 3^2 + 7 ^2$

$58 = 13 + 14 + 15 + 16$ which makes it a trapezoidal number, see here.

$58 = 2 + 3 + 5 + 7 + 11 + 13 + 17$ which means it is the sum of the first $7$ primes, see here.

$58$ is a hendecagonal number.  A hendecagonal number is the equivalent of a triangular number but for a shape that has 11 sides. The first 11 of the 11-gonal or hendecagonal numbers is as follows:
$0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415$

This series can also be generated by writing the natural numbers in a triangular spiral and then reading off the line that starts 0, 1, 11,

..................36
................37..35
..............38..15..34
............39..16..14..33
..........40..17.. 3..13..32
........41..18.. 4.. 2..12..31
......42..19.. 5.. 0.. 1..11..30..58
....43..20.. 6.. 7.. 8.. 9..10..29..57
..44..21..22..23..24..25..26..27..28..56
45..46..47..48..49..50..51..52..53..54..55

## Tuesday, 26 February 2013

### TUESDAY, 26 FEBRURAY 2013

Today is the $57^{th}$ day of the year.

$57 = 3 \times 19$

$57$ is the eighth member of the Tribonacci Numbers, see A000213. Tribonacci Numbers are a natural extension to the Fibonacci Numbers where, instead of the last two numbers being added together, the last three numbers are added together giving a definition of:
$a(n) = a(n - 1) + a(n - 2) + a(n - 3) with a(0) = a(1) = a(2) = 1$
This gives a sequence whose first seventeen numbers are:
$1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, 2209, 4063, 7473$
As you will observe, this sequence gets large very quickly.

## Monday, 25 February 2013

### MONDAY, 25 FEBRUARY 2013

Today is the $56^{th}$ day of the year.

$56 = 2^3 \times 7$

$56$ is the number of times that $11$ can be partitioned. A partition of a number is the number of different ways that the number can be written as a sum of integers where the order is not significant. Thus $11$ has the following partitions:

1. $11$
2. $10 + 1$
3. $9 + 2$
4. $8 + 3$
5. $7 + 4$
6. $6 + 5$
7. $9 + 1 + 1$
8. $8 + 2 + 1$
9. $7 + 3 + 1$
10. $7 + 2 + 2$
11. $6 + 4 + 1$
12. $6 + 3 + 2$
13. $5 + 5 + 1$
14. $5 + 4 + 2$
15. $5 + 3 + 3$
16. $4 + 4 + 3$
17. $8 + 1 + 1 + 1$
18. $7 + 2 + 1 + 1$
19. $6 + 3 + 1 + 1$
20. $6 + 2 + 2 + 1$
21. $5 + 4 + 1 + 1$
22. $5 + 3 + 2 + 1$
23. $5 + 2 + 2 + 2$
24. $4 + 4 + 2 + 1$
25. $4 + 3 + 3 + 1$
26. $4 + 3 + 2 + 2$
27. $3 + 3 + 3 + 2$
28. $7 + 1 + 1 + 1 + 1$
29. $6 + 2 + 1 + 1 + 1$
30. $5 + 3 + 1 + 1 + 1$
31. $5 + 2 + 2 + 1 + 1$
32. $4 + 4 + 1 + 1 + 1$
33. $4 + 3 + 2 + 1 + 1$
34. $4 + 2 + 2 + 2 + 1$
35. $3 + 3 + 3 + 1 + 1$
36. $3 + 3 + 2 + 2 + 1$
37. $3 + 2 + 2 + 2 + 2$
38. $6 + 1 + 1 + 1 + 1 + 1$
39. $5 + 2 + 1 + 1 + 1 + 1$
40. $4 + 3 + 1 + 1 + 1 + 1$
41. $4 + 2 + 2 + 1 + 1 + 1$
42. $3 + 3 + 2 + 1 + 1 + 1$
43. $3 + 2 + 2 + 2 + 1 + 1$
44. $2 + 2 + 2 + 2 + 2 + 1$
45. $5 + 1 + 1 + 1 + 1 + 1 + 1$
46. $4 + 2 + 1 + 1 + 1 + 1 + 1$
47. $3 + 3 + 1 + 1 + 1 + 1 + 1$
48. $3 + 2 + 2 + 1 + 1 + 1 + 1$
49. $2 + 2 + 2 + 2 + 1 + 1 + 1$
50. $4 + 1 + 1 + 1 + 1 + 1 + 1 + 1$
51. $3 + 2 + 1 + 1 + 1 + 1 + 1 + 1$
52. $2 + 2 + 2 + 1 + 1 + 1 + 1 + 1$
53. $3 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1$
54. $2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1$
55. $2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1$
56. $1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1$

## Sunday, 24 February 2013

### SUNDAY, 24 FEBRUARY 2013

Today is the $55^{th}$ day of the year.

$55 = 5 \times 11$

$55$ is a member of the Fibonacci Sequence, is a Triangular Number and is in Pascal's Triangle. Actually, the fact that it is a triangular number implies that it is in Pascal's triangle. This is because, from the third row of the triangle down, the third number in each row is the next member of the triangular number sequence, see here for an illustration.

Slightly less trivially, $55$ is a member of the Toothpick Sequence which is nicely illustrated here or there is an animated version here. To use this animation, leave everything as it is and just click the Next button. When you see the value 10 in the field labelled N: then there are 55 toothpicks in the diagram.

## Saturday, 23 February 2013

### SATURDAY, 23 FEBRUARY 2013

Today is the $54^{th}$ day of the year.

$54 = 2 \times 3^3$

There are some prime numbers that are one greater than a square number. $2,917$ is one such number, see A002496. $2,917 = 2,916 + 1 = 54^2 + 1$, thus $54$ is one of the sequence of numbers $n$ such that $n^2 + 1$ is prime, see A005574. $n$ could be thought of as a Near Root Prime.

It is interesting to speculate on whether there is a number that is in this sequence, let's call it $p$, that when one is added to its square is also a member of this sequence i.e is there a number $p$ such that $p^2 + 1$ is a Near Root Prime. (cf Cunningham chains).

It is fairly easy to show that no such number exists. Assume that $p$ is a Near Root Prime then we know by definition that $p^2 + 1$ is prime. All primes greater than 2 are odd, therefore $p^2 + 1$ is odd. This means that $p^2$ is even which imples that $p$ is even assuming $p$ is greater than $1$. Inspection of A005574 bears this out.
If all the members of the Near Root Prime sequence are even , except the first member, then $p^2 + 1$ cannot be a Near Root Prime because it is odd.

## Friday, 22 February 2013

### FRIDAY, 22 FEBRUARY 2013

Today is the $53^{rd}$ day of the year.

$53$ is the $16^{th}$ prime number.

$53$ is also a Sophie Germain prime. Any prime, p, is a Sophie Germain prime if $2p + 1$ is also a prime. Since $107$ is a prime, the $28^{th}$, then $53$ is a Sophie Germain prime.

The first $23$ Sophie Germain primes are:
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953

### THURSDAY, 23 FEBRUARY 2013

Today is the $52^{nd}$ day of the year.

$52 = 2^2 \times 13$

$52$ is the $5^{th}$ Bell Number.

The $n^{th}$ Bell Number is the number of ways you can split up a set containing n elements. The first non-trivial example is when $n = 3$. Assume that we have a set that contains three elements 1, 2 and 3, then the following five partitions are the only distinct divisions of that set:
{1, 2, 3}
{{1} {2, 3}}
{{2} {1, 3}}
{{3} {1, 2}}
{{1} {2} {3}}

$52$ is the $5^{th}$ Bell Number and the $52$ distinct partitions are:
1. {1, 2, 3, 4, 5}
2. {{1}, {2, 3, 4, 5}}
3. {{1}, {2, 3, 4, 5}}
4. {{3}, {1, 2, 4, 5}}
5. {{4}, {1, 2, 3, 5}}
6. {{5}, {1, 2, 3, 4}}
7. {{1, 2} {3, 4, 5}}
8. {{2, 3} {1, 4, 5}}
9. {{3, 4} {1, 2, 5}}
10. {{4, 5} {1, 2, 3}}
11. {{5, 1} {2, 3, 4}}
12. {{1, 3} {2, 4, 5}}
13. {{2, 4} {1, 3, 5}}
14. {{3, 5} {1, 2, 4}}
15. {{4, 1} {2, 3, 5}}
16. {{5, 2} {1, 3, 4}}
17. {{1} {2} {3, 4, 5}}
18. {{1} {3} {2, 4, 5}}
19. {{1} {4} {2, 3, 5}}
20. {{1} {5} {2, 3, 4}}
21. {{2} {3} {1, 4, 5}}
22. {{2} {4} {1, 3, 5}}
23. {{2} {5} {1, 2, 4}}
24. {{3} {4} {1, 2, 5}}
25. {{3} {5} {1, 2, 4}}
26. {{4} {5} {1, 2, 3}}
27. {{1} {2, 3} {4, 5}}
28. {{1} {2, 4} {3, 5}}
29. {{1} {2, 5} {3, 4}}
30. {{2} {1, 3} {4, 5}}
31. {{2} {1, 4} {3, 5}}
32. {{2} {1, 5} {3, 4}}
33. {{3} {1, 2} {4, 5}}
34. {{3} {1, 4} {2, 5}}
35. {{3} {1, 5} {2, 4}}
36. {{4} {1, 2} {3, 5}}
37. {{4} {1, 3} {2, 5}}
38. {{4} {1, 5} {2, 3}}
39. {{5} {1, 2} {3, 4}}
40. {{5} {1, 3} {2, 4}}
41. {{5} {1, 4} {2, 3}}
42. {{1} {2} {3} {4, 5}}
43. {{1} {2} {4} {3, 5}}
44. {{1} {2} {5} {3, 4}}
45. {{1} {3} {4} {2, 5}}
46. {{1} {3} {5} {2, 4}}
47. {{1} {4} {5} {2, 3}}
48. {{2} {3} {4} {1, 5}}
49. {{2} {3} {5} {1, 4}}
50. {{2} {4} {5} {1, 3}}
51. {{3} {4} {5} {1, 2}}
52. {{1} {2} {3} {4} {5}}

## Wednesday, 20 February 2013

### WEDNESDAY, 20 FEBRUARY 2013

Today is the $51^{st}$ day of the year.

$51 = 3 \times 17$

$51$ is the sixth Motzkin Number. A Motzkin number is the number of different ways of drawing non-intersecting chords on a circle between n points on the circle's circumference. Here I will unashamedly copy the illustration from Wikipedia which shows that when $n = 4$ the Motzkin number is $9$ i.e. there are $9$ different ways to draw lines connecting $4$ points, not necessarily all of the four points, on the circumference of a circle without the lines intersecting.

The first 21 Motzkin numbers are:
1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559

Like the Fibonacci sequence, this sequence has a relationship that links the next number in the sequence to the previous two numbers. This relation is
$M_{n+1}=\frac{2n+3}{n+3}M_n+\frac{3n}{n+3}M_{n-1}$.
Thus
$M_6 = \frac{2.5+3}{5+3}M_5+\frac{3.5}{5+3}M_4$
knowing that $M_5 = 21$ and $M_4 = 9$ from the list above we get
$M_6 = \frac{13}{8}.21+\frac{15}{8}.9$
$M_6 = \frac {1} {8} (13. 21 + 15 . 9)$
$M_6 = \frac {1} {8} (273 + 135)$
$M_6 = \frac {1} {8} . 408$
Thus
$M_6 = 51$

## Tuesday, 19 February 2013

### TUESDAY, 19 FEBRUARY 2013

Today is the $50^{th}$ day of the year.

$50$ is an odious number and a Harshad or Niven number.

An odious number is any number that has an odd number of ones in its binary expansion.
$50_{10} = 110010_{2}$ so it has three ones in its binary form and is therefore odious.
The first 31 odious numbers are:
1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61

Those numbers that are not odious are Evil Numbers.
The first 30 Evil Numbers are:
0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58

A Harshad or Niven Number is any number that is divisible by the sum of its digits.
The sum of the digits of $50$ is $5 + 0 = 5$ and, clearly, $50$ is divisible by $5$.
The first 30 Harshad numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81

## Monday, 18 February 2013

### MONDAY, 18 FEBRUARY 2013

Today is $49^{th}$ day of the year.

$49$ is a perfect square.

$49$ is also a lucky number.

The best description of lucky numbers that I have come across is from Ivars Peterson's MathTrak blog which I quote from below:

Hunting for prime numbers, those evenly divisible only by themselves and 1, requires a sieve to separate them from the rest. For example, the sieve of Eratosthenes, named for a Greek mathematician of the third century B.C., generates a list of prime numbers by the process of elimination.
To find all prime numbers less than, say, 100, the hunter writes down all the integers from 2 to 100 in order (1 doesn't count as a prime). First, 2 is circled, and all multiples of 2 (4, 6, 8, and so on) are struck from the list. That eliminates composite numbers that have 2 as a factor. The next unmarked number is 3. That number is circled, and all multiples of 3 are crossed out. The number 4 is already crossed out, and its multiples have also been eliminated. Five is the next unmarked integer. The procedure continues in this way until only prime numbers are left on the list. Though the sieving process is slow and tedious, it can be continued to infinity to identify every prime number.
Other types of sieves isolate different sequences of numbers. Around 1955, the mathematician Stanislaw Ulam (1909-1984) identified a particular sequence made up of what he called "lucky numbers," and mathematicians have been playing with them ever since.
Starting with a list of integers, including 1, the first step is to cross out every second number: 2, 4, 6, 8, and so on, leaving only the odd integers. The second integer not crossed out is 3. Cross out every third number not yet eliminated. This gets rid of 5, 11, 17, 23, and so on. The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39, ... Now, the fourth number from the beginning is 9. Cross out every ninth number not yet eliminated, starting with 27.
This particular sieving process yields certain numbers that permanently escape getting killed. That's why Ulam called them "lucky." See the table below for a list of lucky numbers less than 200.
1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163 169 171 189 193 195

These lucky numbers should not be confused with Euler's Lucky Numbers, see http://oeis.org/A014556.

For those that are interested, the $66$ days of this year that are related to a lucky number are:
 Lucky Number Date 1 Tuesday 1 January 3 Thursday 3 January 7 Monday 7 January 9 Wednesday 9 January 13 Sunday 13 January 15 Tuesday 15 January 21 Monday 21 January 25 Friday 25 January 31 Thursday 31 January 33 Saturday 2 February 37 Wednesday 6 February 43 Tuesday 12 February 49 Monday 18 February 51 Wednesday 20 February 63 Monday 4 March 67 Friday 8 March 69 Sunday 10 March 73 Thursday 14 March 75 Saturday 16 March 79 Wednesday 20 March 87 Thursday 28 March 93 Wednesday 3 April 99 Tuesday 9 April 105 Monday 15 April 111 Sunday 21 April 115 Thursday 25 April 127 Tuesday 7 May 129 Thursday 9 May 133 Monday 13 May 135 Wednesday 15 May 141 Tuesday 21 May 151 Friday 31 May 159 Saturday 8 June 163 Wednesday 12 June 169 Tuesday 18 June 171 Thursday 20 June 189 Monday 8 July 193 Friday 12 July 195 Sunday 14 July 201 Saturday 20 July 205 Wednesday 24 July 211 Tuesday 30 July 219 Wednesday 7 August 223 Sunday 11 August 231 Monday 19 August 235 Friday 23 August 237 Sunday 25 August 241 Thursday 29 August 259 Monday 16 September 261 Wednesday 18 September 267 Tuesday 24 September 273 Monday 30 September 283 Thursday 10 October 285 Saturday 12 October 289 Wednesday 16 October 297 Thursday 24 October 303 Wednesday 30 October 307 Sunday 3 November 319 Friday 15 November 321 Sunday 17 November 327 Saturday 23 November 331 Wednesday 27 November 339 Thursday 5 December 349 Sunday 15 December 357 Monday 23 December 361 Friday 27 December

## Sunday, 17 February 2013

### SUNDAY, 17 FEBRUARY 2013

Today is $48^{th}$ day of the year.

$48$ is the double factorial of $6$, written $6!!$

The value of the double factorial of a number $n$ is defined as follows:
If $n = -1$ or $n = 0$ then $n!! = 1$ otherwise $n!! = n.(n - 2)!!$

Given $n = 6$ then $6!! = 6.4!! = 6.4.2!! = 6.4.2.0!! = 6.4.2.1 = 48$
However, if $n = 5$ then $5!! = 5.3!! = 5.3.1!! = 5.3.1 = 15$
Thus, $6!!.5!! = 6.4.2.1.5.3.1 = 6.5.4.3.2.1 = 6!$
or, equivalently:
$$n!!.(n - 1)!! = n!$$

## Saturday, 16 February 2013

### SATURDAY, 16 FEBRUARY 2013

Today is $47^{th}$ day of the year.

47 is a prime number.

47 is the $8^{th}$ Lucas Number, see http://oeis.org/A000032

The Lucas Number are like the Fibonacci numbers in that the $n^{th}$ number is the sum of the previous two numbers, i.e. the $(n-1)^{th}$ and the  $(n-2)^{th}$. In the case of the Fibonacci numbers the first two numbers are 0 and 1 giving the sequence
0, 1, 1, 2, 3,  5,  8, 13, 21, 34,  55,  89, 144, 233, 377, ...
whereas the Lucas Numbers start with 2 and 1 giving the following sequence:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, ...
You may be able spot an interesting link between these two series; the $n^{th}$ Lucas Number is equal to the sum of the  $(n-1)^{th}$ and the  $(n+1)^{th}$ Fibonacci Number i.e the eighth Lucas Number, 47 , is equal to the seventh Fibonacci Number, 13, plus the ninth Fibonacci Number, 34.
We could write this as:
$$L_{n}=F_{n-1}+F_{n+1}$$
Using the same nomenclature, Wikipedia informs us that there are a number of other identities:
\begin{align}L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1}\end{align}
\begin{align}L_n^2 = 5 F_n^2 + 4 (-1)^n\end{align}
\begin{align}F_{2n} = L_n F_n\end{align}
\begin{align}F_n = {L_{n-1}+L_{n+1} \over 5}\end{align}
Equation 1
With $m=3$ and $n=5$ then $L_8 = L_4.F_5 + L_3.F_4 = 7.5 + 4.3 = 35 + 12 = 47$

Equation 2
With $n = 8$ then $L_8^2 = 5.F_8^2 + 4.(-1)^8 = 5.{21}^2 + 4 = 5.441 + 4 = 2,209 = 47^2$

Equation 3
With $n = 8$ then $F_{16} = L_8.F_8 = 47.21 = 987$

Equation 4
With $n = 7$ then $F_7 = {L_6 + L_8 \over 5} = {18 + 47 \over 5} = {65 \over 5} = 13$

## Friday, 15 February 2013

### FRIDAY, 15 FEBRUARY 2013

Today is 46th day of the year.

Take one of the pizzas mentioned in yesterdays post and make nine straight cuts with a pizza cutter. What is the maximum number of pieces of pizza (not necessarily the same size) that can be created with this approach? Not surprisingly it 46. An illustration of the first few members of this sequence, the Lazy Caterer's Sequence, can  be found at http://oeis.org/A000124/a000124.gif. It looks like this:

## Thursday, 14 February 2013

### THURSDAY, 14 FEBRUARY 2013

Today is the 45th day of the year.

If you had a choice of two pizza toppings from a selection of 10 then there would be  45 different combinations from which to choose.

Let us assume that the available toppings are Pepperoni, Cheese, Sausage, Mushrooms, Pineapple, Bacon, Ham, Shrimp, Onions and Green Peppers, then the daily toppings could have been:

 Date Topping 1 Topping 2 Tuesday 01 January Pepperoni Cheese Wednesday 02 January Pepperoni Sausage Thursday 03 January Pepperoni Mushrooms Friday 04 January Pepperoni Pineapple Saturday 05 January Pepperoni Bacon Sunday 06 January Pepperoni Ham Monday 07 January Pepperoni Shrimp Tuesday 08 January Pepperoni Onions Wednesday 09 January Pepperoni Green Peppers Thursday 10 January Cheese Sausage Friday 11 January Cheese Mushrooms Saturday 12 January Cheese Pineapple Sunday 13 January Cheese Bacon Monday 14 January Cheese Ham Tuesday 15 January Cheese Shrimp Wednesday 16 January Cheese Onions Thursday 17 January Cheese Green Peppers Friday 18 January Sausage Mushrooms Saturday 19 January Sausage Pineapple Sunday 20 January Sausage Bacon Monday 21 January Sausage Ham Tuesday 22 January Sausage Shrimp Wednesday 23 January Sausage Onions Thursday 24 January Sausage Green Peppers Friday 25 January Mushrooms Pineapple Saturday 26 January Mushrooms Bacon Sunday 27 January Mushrooms Ham Monday 28 January Mushrooms Shrimp Tuesday 29 January Mushrooms Onions Wednesday 30 January Mushrooms Green Peppers Thursday 31 January Pineapple Bacon Friday 01 February Pineapple Ham Saturday 02 February Pineapple Shrimp Sunday 03 February Pineapple Onions Monday 04 February Pineapple Green Peppers Tuesday 05 February Bacon Ham Wednesday 06 February Bacon Shrimp Thursday 07 February Bacon Onions Friday 08 February Bacon Green Peppers Saturday 09 February Ham Shrimp Sunday 10 February Ham Onions Monday 11 February Ham Green Peppers Tuesday 12 February Shrimp Onions Wednesday 13 February Shrimp Green Peppers Thursday 14 February Onions Green Peppers

## Wednesday, 13 February 2013

### WEDNESDAY, 13 FEBRUARY 2013

Today is $44^{th}$  day of the year.
$$44^{16} + 1 = 197,352,587,024,076,973,231,046,657$$
$197,352,587,024,076,973,231,046,657$ is a prime number.