> Look at the identity: 4 = 2 × 2.
Conclusion: 4 is the smallest composite number (a composite number can be written as the product of smaller positive integers). Number 4 is a perfect square; it is also a power of 2.
> Quadrilaterals (4-sided polygons) include many common shapes, like rectangles, parallelograms, etc. Can you give more examples?
> In a plane there are 4 directions: up, down, left and right.
> A tetrahedron (4-faced polyhedron) has 4 vertices, 6 edges and 4 faces.
A tetrahedron – also called a triangular pyramid, is the simplest polyhedron (in terms of number of faces is smallest).
There is a favourite topic in science fiction: 4-dimensional space.
> Number 4 inspired the imagination of mathematicians and their enthusiastic exploration: and yield fruitful results: the most famous are “four-colours theorem” and the “sum-of-four-squares” [this item is a bit hard].
About Four-colours Theorem: On a plane or a sphere, it suffice to employ only four colours to seperate all neighbouring regions apart.
About Sum-of-four-squares theorem: Any integer can be written as the sum of at most four squares.
For example: 5 = 22 + 12, 25 = 42 + 32 [As Sum of two perfect squares]
7 = 22 + 12 + 12 + 12 [As Sum of four perfect squares]
> If we list all powers of 4: 4, 16, 64, 256, 1024, 4096, etc.
Have you noticed: the last digit (units-place) is either 4 or 6.
By the way, the complimentary number of 4 is 6, and 4 × 6 = 24.
Conclusion: ________________ (The reader may summarize by himself).
> The sum of four consecutive integers is an even number, however, it is NEVER a multiple of 4. For example: 1 + 2 + 3 + 4 = 10.
Since ten resembles “perfect”, so as number 4 is loved in some cultures (for example, the Pythagorean scholars in ancient Greece).
Last Spring, Jonah’s Math Corner hosted a column called “Prime Fun Numbers”. Lots of feedback were received that the column were welcomed by math kids and their parents as well.
That’s why we kick off 2017 with this same column, and updated contents.
The first post will be on number 17 (seventeen). What have you thought of when hear 17? Stay tuned ..
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 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).
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），2，4，8，16， …
In such a sequence, the sum of consecutive items starting from 1 is always one less than the next item, e.g., 1 = 2-1， 1+2 = 4-1， 1+2+4 = 8 -1，etc.
> 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!
