*Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. *

**Leonhard Euler**

Numbers play a significant role in our daily lives, from the time we wake up and check the clock to the time we go to bed and count sheep. However, despite their prevalence, we often take them for granted and overlook their fascinating mathematical properties. In this blog post, we will explore some of the intriguing properties of familiar numbers that may have gone unnoticed.

- 2 is the only prime number without an
**e**in its name. - The only number equal to its factorial since 2! = 2.
- 2 is the only natural number satisfying the property 2 + 2 = 2 × 2.
- For any convex polyhedron, the number of faces plus the number of vertices, minus the number of edges, is 2:
*V*‒*E*+*F*= 2. - The sum of the reciprocals of the factors of any perfect number is equal to 2.
- 2 is the base of the binary number system.
- 2 related items are often called a pair, and words like dual, duel, couple, twin, and double emphasize the significance of the number two.
- We can express
*π*as an infinite product containing only 2 and its reciprocal ½.

- 73 is the 21st prime number. Its mirror, 37, is the 12th prime number (which is the mirror of 21). The sum of the digits in both numbers is 10, and their product is 21.
- If you add 100 to both 37 and 73, you get 137 and 173, respectively, which are both prime numbers.
- The number of days in an ordinary year is 365, the product of the two primes 5 and 73.
- The alphabetic value of the word
**NUMBER**(= 2 + 5 + 13 + 14 + 18 + 21) is 73. - The number 73 is the only
**EMIRP**that is one less than the double of its reversal: 73 = 2 × 37 ⎼ 1. - If you strip off the six triangular points from the 73-circle hexagram, you are left with a hexagon of 37 circles.

- All three-digit repunits are divisible by 37.
- If the multiples of 37 be mirrored and then separated by a zero, they will be another multiple of 37. For example, 518 and 80105.
- Any multiple of 37 that has a three-digit repunit inserted will generate another multiple of 37. Examples of such numbers include 30007 and 78884.
- From a three-digit number
*abc*, form two more numbers*cab*and*bca*, in that order. Add all three numbers together. The resulting number is always divisible by 37. From 456, create 645 and 564. The sum of them is divisible by 37. - The chances of a wedding for the
**Sultan’s Dowry Problem**are about 37%!

- 18 is the only
**two-digit**number that is twice the sum of its digits. - 18 is the only number where the sum of its digits 1 + 8 = 9 is equal to half of itself (18/2 = 9).
- The number 18 = 31 − 13 = 97 − 79 is the smallest positive difference between an
**EMIRP**and its reverse. - The cube of 18 and the fourth power of 18 together use each digit exactly once: 18
^{3}= 5832 and 18^{4}= 104976. *The*18-*point problem*is to place a sequence of 18 points on an interval so that the first two points are in separate halves of the interval, the first three points are in separate thirds of the interval, the first four points are in separate fourths of the interval, and so on.

- The sum of the digits of 81 (= 8 + 1 = 9) when multiplied by its reversed self, yields the original number (9 × 9 = 81).
- 81 is the smallest square such that the sum of its divisors, 1 + 3 + 9 + 27 + 81 = 121 = 11
^{2}, is also a square. - It is the only two-digit number that is the square of the sum of its digits: 81 = (8 + 1)
^{2}. - In the decimal expansion of
**1/81**= 0.**012345679012345679**. . . , each digit appears (in order)**except 8**. - There are 81 rectangles on a Shogi board (Japanese chess).

- 69 days = 6 days 9 weeks
- The sum of the divisors of 69 is equal to its reversal: 96 = 1 + 3 + 23 + 69.
- The numbers 69
^{2}=**4761**and 69^{3}=**328509**together contain each of the digits exactly once. - The alphabetical value of the letters in 69’s Roman numeral LXIX is 12 + 24 + 9 + 24 = 69.
- 69 is equal to 105 octal, while 105 is equal to 69 hexadecimal.
- The numeral 69 looks the same when rotated by 180º.

- 96 is the second smallest number with six prime factors: 96 = 2 × 2 × 2 × 2 × 2 × 3.
- A number with one-eighth the number of divisors of the original number.
- There are 96 spokes on a daisy wheel printer.
- The Courier game is an old variant of chess played on a board with 96 squares.
- 96 is an untouchable number.
- 96 appears the same when turned upside down.
- Skilling’s figure, a degenerate uniform polyhedron, has a Euler characteristic χ = ‒ 96.
- 96 is 88 in base 11.

- 23 is the smallest prime number with consecutive digits.
- The smallest prime whose reversal is a power: 32 = 2
^{5}. - Using congruence, we can write 23 ≡ 2 (mod 3) and 23 ≡ 3 (mod 2).
- 23 differs from its reversal 32 by 3
^{2}. - There are 23 definitions in Book I of Euclid’s
.*Elements* **W**is the 23rd letter of the modern English alphabet. Have you ever noticed that it has 2 points down and 3 points up?- In a room of just 23 people, a greater than 50% chance exists that two of the people will share a common birthday.
- Representation using its digits: 23 = 1! + (2! + 2!) + (3! + 3! + 3!).
- The sum of squares of its digits 2
^{2}+ 3^{2}= 4 + 9 = 13 is also a prime number. - If we add the square of the digits of the number 23 successively, we eventually reach 1: 2
^{2}+ 3^{2}= 13; 1^{2}+ 3^{2}= 10; 1^{2}+ 0^{2}= 1.

- The number 32 is (possibly) the highest power of 2 with all prime digits.
- As a sum of products: 32 = 1 × 2
**+**1 × 2 × 3**+**1 × 2 × 3 × 4. - As a sum of powers: 32 = 1
^{1}+ 2^{2}+ 3^{3} - There are 32 fluid ounces to the quart, 32 gills to the gallon, 32 dry quarts to the bushel, and 32 ells to the bolt.
- The freezing point of pure water at sea level is 32˚F.
- The acceleration of gravity on the surface of the Earth is about 32 feet per second per second.

- The number 63 is the smallest number
*n*whose Roman numeral has alphabetic value*n*. Indeed, the value of LXIII is 12 + 24 + 9 + 9 + 9 = 63. - It is the sum of the powers of 2 from 0 to 5: 63 = 2
^{0}+ 2^{1}+ 2^{2}+ 2^{3}+ 2^{4}+ 2^{5} - There are 63 posets with 5 unlabelled elements.
- A barrel that holds 63 gallons is called a
*hogshead*.

- 36 is the number of degrees in the interior angle of each tip of a regular pentagram.
- The
**36****officers’ problem**(proposed by Euler) is a mathematical puzzle with*no solution*. - It is the number of possible outcomes in the roll of two distinct dice.
- 36 (numerals 0–9 and the letters A‒Z) is the number of alpha-numeric keys.
- The
*truncated cube*and the*truncated octahedron*are Archimedean solids with 36 edges. - 36 is the number of inches in a yard (3 feet).
- It is 3 dozen, or a quarter of a gross.

- 13 is the smallest EMIRP.
- A
is a group of 13.*baker’s dozen* - A reflectional type property: 13
^{2}= 169 and its reversal 31^{2}= 961. - There are 13 Archimedean solids.
- Three planes can cut a donut into a maximum of 13 parts.
- The sum of primes up to 13 is equal to the 13th prime.
- Consider the primes up to 13. While dividing 13 by each of these primes individually, the sum of the remainders is 13.
- The fear of the number 13 is called triskaidekaphobia.
- 13 is the number of cards in one suit out of a deck of fifty two cards.
- The dice game Yahtzee consists of 13 rounds.
- 13 is ELEVEN + TWO = TWELVE + ONE. Can you solve this anagrammatically?

- 31 is the number of days in seven months.
- It is the sum of the first two primes raised to themselves: 2
^{2}+ 3^{3}= 31. - The reverse of 31 is 13 which is also a prime. So 31 is an EMIRP.
- 31 is the smallest prime that is the sum of the divisors of two different numbers, 16 and 25.
- 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.
- The cube root of 31 is the value of pi correct to four significant figures (∛31 = 3.1413…).

- 17 is the (1 × 7)
^{th}prime. - It can be written as the powers of first two primes: 17 = 2
^{3}+ 3^{2}. - 17 is an EMIRP.
- It is the smallest prime that is the sum of consecutive primes: 17 = 2 + 3 + 5 + 7.
- 17 is the smallest number of givens (prefilled squares) that a Sudoku puzzle can have.
- It is equal to the sum of digits of its cube: 17
^{3}= 4913 and 4 + 9 + 1 + 3 = 17. - 17 is the smallest prime sandwiched between two non-square free numbers (4
^{2}< 17 < 2 × 3^{2}). - There are 17 standard-form types of quadratic surfaces (quadrics).
- 17 is the number of different ways that a wallpaper design can repeat.
- The fear of the number 17 is called
.*heptadecaphobia*

- Conway’s constant is an algebraic number of degree 71.
- 71 is the largest number known whose square is a factorial plus one: 71
^{2}= 7! + 1! - The digits of 71
^{3}= 357911 are the odd numbers from 3 to 11 in sequence. - The number 71 divides the sum of the primes that are less than 71.
- The number (71
^{71}‒ 71!)/71 is prime. - 71 is the largest number known whose square is one more than a factorial: 71
^{2}= 7! + 1. - When a few zeros are inserted between the digits 7 and 1, they produce primes: 701, 7001, 70001, and 700001.

These are just a few examples of the fascinating mathematical properties that familiar numbers can possess. By exploring these properties, we can gain a deeper appreciation for the numbers we encounter every day and perhaps even find new ways to use them in our lives.

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call **“****Math1089 – Mathematics for All!**“.