## Number 17: Mysterious and Special

Let’s start by some facts regarding 17 as a prime number. A prime is a whole number who has only two divisors: 1 and itself. 2 is the smallest prime. 3, 5, 7 .. are primes too. 6 = 2 x 3 is not a prime: 6 has four factors: 1, 2, 3, 6.

> 17 is the 7th prime number.

2, 3, 5, 7, 11, 13, 17 : Yeah, the seventh prime is 17.

> 17 and 19 are twin primes: twin primes are two consecutive odd numbers who are both primes.

> This year is 2017. Both 17 and 2017 are prime numbers. But since 117 = 3 × 3 × 13, 2117 = 29 × 73: neither 117 nor 2117 is a prime number.

Here are some classical results to which are linked two big names (both mathematicians) : Fermat (Pierre de Fermat) and Gauss (Carl Friedrich Gauss).

> 17 is a Fermat prime. Fermat numbers are in the form of 2 2 n (meaning: 2 raised to the 2n power) as n = 0, 1, 2, .. .. When setting n = 2, we get the third prime 17.

Carl Friedrich Gauss discovered that heptadecagon (17-sided polygon) is constructable using compass (for drawing circles) and ruler (for drawing lines) only. Any regular polygon with number of sides being a Fermat prime number can be such constructed (as Gauss gave the method of construction and the proof).

17 is everywhere – from Sudoku game to Irregularity Distribution

> (An Interesting Claim about a soduko game) Have you played a sudoku game? It is a puzzle of 9-by-9 (=81) cells, each cell supposed to be filled with a digit chosen from 1 to 9. Some entries of the 81 cells are given prior, then a player’s task is to fill remaining entries

such that each row, each column, and each 3-by-3 smaller cell must contain digits from 1 to 9, each appearing exactly once.  If such conditions are all met, the fill-in is called a (proper) solution.

See above figures for a given sudoku game (with some fixed entries given) and one of solutions.

Regarding the game, there is a claim:

If a sudoku game leads to one unique solution, then a necessary condition is, at least 17 entries must have been given at the very beginning of the game.
(Otherwise, the sudoku game cannot have a unique solution.)

Let us see an example. If there is a 9-by-9 sudoku game where you see only 15 cells filled (with numbers from 1 to 9), then you must be able to work out two or more different solutions (or none at all).

The claim was proved quite recently — in 2012.

> For the problem of Irregularity of (1/n)- distribution events, the solution exists for n<=17.

(Details are to be posted soon)

We conclude with a few trivia.

> (Sum of Squares and Cubes)

17 = 32 + 23 = 42 + 13

That is, 17 can be written as the sum of one perfect square number and one perfect cube number in two different ways. 1st way uses numbers 2 and 3 only; 2nd way uses numbers 1, 2, 3, 4 resp.

Among all that possess the feature (as a prime and can be written into such sums), 17 is the smallest.

> (17 is the least random number)

In a psychology test, when asking to write down a random number between 1 and 20, most people chooses 17 – such got the name “the least random number”.

> (17 – as in Seventeen: in album, name of bands, journals, movies and music)

In countless albums, bands, movies and magazines, journals, you’ll find 17 as part of the title or name. As I did a search, several songs immediately pop up:

• “17” (Sky Ferreira song)
• “17” (Yourcodenameis:Milo song)
• “17 Again”, a song by Eurythmics
• “17 år”, a song by Veronica Maggio
• “17 Crimes”, a song by AFI
• “17 Days”, a song by Prince

The reason is mysterious, but that age 17 as youth curiosity and restlessness may have played a role.

>(number morphing and factoring)

17 x 71 = 1207 -> 2017

Reverse the procedure: from 2017 shuffle the digits we can get 1207 — we call that number morphing, and then we are to factor 1207 to 17 x 71.
Wow, That’s Seventeen. Apart from something deep (like Gauss’s results), we are here Just for Fun!

## “Three”: Tripod, Three Sides make a Triangle

“Three”    查看中文： “3” 三人成众

Three sounds pretty close to “tree”, there are lots of trees in a forest.
Have you heard the word “tripod”? Here “tri-” means three.

> The minimal property for number 3: a polygon has at least 3 sides, the smallest odd prime is 3. The first and the smallest triangle number is also 3 (number 3 = 1 + 2).

If not counting “1”, then number 3 is the smallest odd number.

Oddly enough, “3” also means “many” in some cultures.

> All integers are divided into 3 categories: positive integers, zero, and negative integers.

> 32 = 3 X 3 = 9, 23 = 2 X 2 X 2 = 8; 9 and 8 are the largest and the second largest digits in the decimal (base-10) system.

>

To test whether a number is divisible by 3 (has factor 3), it is enough to add all the digits of the number, and check weather the sum of digits has factor 3.

Example: 15 [1+5=6], 6 has factor 3, so 15 is divisible by 3.

121 (1+2+1=4), 4 does not contain factor 3, therefore 121 is not divisible by 3.

> The sum of any three consecutive integers is divisible by 3.

> The 3-4-5 triangle is the smallest right triangle with all three integer side lengths.

This fact is also interesting: if a right triangle has three sides of integer lengths, then at least one side length is a multiple of 3.

Examples: we list some right triangles with integer side lengths:
3-4-5, 5-12-13, 8-15-17, 7-24-25,
in which 3, 12, 15, 24, .. are the multiples of 3.

> When we say the most, we mean a fraction of total: the fraction may be (2/3) (two-thirds) or (3/4) (three quarters).

## “Two” – In Pairs

“Two”

Whether you like it or not, number “2” is fairly important in math.

> Any number that can be divisible by 2 (has a factor of 2) is called an even number. Otherwise it is an odd number.

> 2 is the smallest prime number. (A prime number is a natural number that cannot be written as the product of even smaller numbers).

2 has two divisors: 1 and 2. Any prime number has two divisors.

> Look at the identity.

x + x = 2x, (x) (x) = x2

where x2 is read as “x square”.

Conclusion: we meet number “2” quite often in both arithmetic and algebra.

> 2 is the base for binary system (Carry one to next higher place whenever we reach two). The numbers in a binary system are 0, 1, 10, 11, 100, 101, … .

> The numbers in power of 2, when written in decimal system (the usual system) are:

1），24816

In such a sequence, the sum of consecutive items starting from 1 is always one less than the next item, e.g., 1 = 2-11+2 = 4-11+2+4 = 8 -1etc.

> Look at this equation on the sum:

1 + 1/2 + 1/(22) + 1/(23) + … = 2

Conclusion: what would you say? _______________

> (Algebric) Squares are the powers when the exponent is 2. We calculate the area of a square shape by squaring side of that shape.

From Pythagorean Theorem: in a right triangle, the sum of squares of the two sides equals the square of the length of hypotenuse.

> The binomial power (a+b)n is studied in early classic algebra and is a classic topic in high school algebra. (We do not have 2 here, however, a and b are two items).

Expansion this into polynomial, one tool is Pascal triangles. [This entry is a bit deeper.]

> When thinking of “two”, think of being in pairs, and think of even and odd!