Prime Factors: Part 6

Prime Factors:

The Chart

Finally, Down to the basic formulas.  In this chart, or more specifically, diagram, one can find all the parts of the formulas used to determine whether a large number is divisible by a particular prime number.

Usually, a large number is reduced first by dividing by 2, 3, 5 and 7.  Then, an estimate of the square root of the remaining number gives the upper limit for searching for prime factors.  The chart works for the double digit primes as well as for "03" and "07."

The "rule" of the "sum of the digits" for the 3 can be derived from the X3 quadrant, assuming the format of "03."  Likewise, plugging "07" into the X7 quadrant gives the "trick" or "rule" for finding out if a number is divisible by that prime number.

To review:

For divisibility

 by 2: Look for the even number on the end.

 by 3: The sum of the digits will be divisible by 3.

 by 5: Look for the 5 (or 0, but see 2) at the end of the number.

 by 7: Using the formula for the X7 primes: For "07"10x+n = x-(n*(0+2))

          That is, for divisibility by 7, for any number "10x+n" take twice n from x until you get to a multiple of 7.

Double digit primes below 100 are on the chart.  Larger primes can be added remembering to put them in the right "family."

101 is in the X1 family, Therefore, plugging it into the formula: 10-(1*10) => K*X1.  Of course, with the zero in the middle, numbers up to 10,000 are easy to spot as multiples of this prime!

8989 => 898-90 = 808. Of course, that choice was influenced by my building the chart.  I put up a reduplicative whole number!

Really big odd numbers always give us a challenge.  Once divisibility by 3 and 5 are eliminated, the chart come into play.  I am going to blindly punch in 7 digits, making sure that the end number is not even or a 5.

7351613.  7+3+5+1+6+1+3 = 26.  So, not divisible by 3.

Let us try 7.

7351613 => 735161-(3*2)=> 735155=> 735155-(5*2)=> 735145=> 7351-(5*2)=> 734-2=> 732. 

732 is not divisible by 7.

What about 11?

7351613 => 735161-3=> 73515-8=> 73507=>7350-7=> 7343=> 431=> 42.  Nope.

So, 13?

7351613 => 735161+(3*4) = 735173 => 73517+(3*4) = 73539 => 7353+(9*4) = 7381 => 738+4 =

742 => 74+(2*4) = 74+8 = 82.  13 does not work. 

Which of the two attempts got closer?  42 is 2 away from 44, while 82 is 4 away from 78.  Perhaps the X1 family offers a better chance.

What of 31, which requires a multiplier of 3, two away from 1.

7351613 => 735161-3*6=> 7350-9*3=> 732-3*3=> 72-1*3 = 69.  69 = 62+7.  Further away!

So, let us first estimate a square root by dividing the seven digits into groups of two starting from the right.

7 35 16 13.  The square root will be a number between 2000 and 3000.  2,500*2,500 = 6,250,000.

So, my guess is that the limit of possibilities lay somewhere around 2750.  Ending in a 3, the number is not an exact square.  A lot of work to go, but I think I've written enough.

So, what do I know?

The search for primes continues past 97.  But, placing all possible primes in the four "families" makes the job a bit easier.  Once the one gets past division by 7, at least without a calculator, having derived a family formula helps to discover larger primes by elimination.

I have not memorized the above chart, but I did use simple math to derive the formulae.  The exercise of my brain has staved off mental deficiencies for at least a few months!

I don't know if finding the largest prime number yet is worth the effort, but perhaps using this chart will help someone in the search.

I know that the chart will help an inquiring mind find proof that 7,351,613 is not prime.  I challenge a reader, whether a friend or an random reader, to send me the prime factor.  You DO have a clue.

  

Prime Factors: Part 3

Prime Numbers:

The Amazing Mr. Five.

Of the prime numbers under 10, only 5 stands alone.  He has no other primes in his family.  As such, recognizing multiples are easy. Just look for a 5 at the end.

This is true, because any number that is divisible by 10 -- that is has a 0 at the end -- divisible by both 2 and 5.  Further determination is made after removing the 0.

For instance:

1234560 is divisible by 2 and 5 (that is, by 10). That removes 5 from the running, at least temporarily.  Here is the progression:

1234560/2*5 = 123456
123456/2    = 61728
61728/2     = 30864
30864/2     = 15432
15432/2     = 7716
7716/2      = 3858
3858/2      = 1929
1929/3      = 543
543/3       = 271

My guess is that 271 is prime. It is one over 270, which has obvious factors.  Further investigation bears me out.

The "Five and Dime" Store.

Being that 5*2 is 10, and inversely 10/2 is 5, multiplying and dividing by 5 is easy. You only need to multiply by 10  and divide by 2 to get any number multiple of 5.

123*5 = 123*10/2 = 1230/2 = 615

Using the decimal point, it works the other way:

(123/2)(10) =61.5*10 = 615

Finally, it is easy to spot numbers divisible by 25 (5^2) and 125 (5^3). This is because 100 and 1000 are multiples of 10.

For 25, there are 3 multiples: 25, 50 and 75.  For example:

123,450 is divisible by 5 and 25. 50 is 2*5*5. So 2, 5, 10 and 25 are factored out.

123,450/2*5 = 12,345
12,345/5    = 6,171
6,171/3     = 2057, an odd number.  Is it prime? Stay tuned.

Another, example:

987,625 is divisible by 125 (5*25=625).  This would also place a 5 on the end of the next factor, raising the power by another 5.

8*125 = 2^3*5^2 = 10^2 = 1000

This leaves 987,000.  Removing the zeros, we then look for the factors, if any, of 987.  3 works, though 9 doesn't.

987/3 = 329.

So, dividing an odd by an odd, we've ended up with an odd number.  Further calculations are needed, but our odd number is within reach.  20*20 is 400, setting a limit.

So, what do I know?

I know that 5 and 10 are closely related by the prime number 2.  And this makes multiplying and dividing by 5 very easy. There are no other prime numbers related to 5, making the possibility of a random number being prime at less than 25%.

Even so, mathematically, there are an infinite number of prime numbers.  This is because "infinity" can not be divided. There are an infinite number of fractions.

Very odd, so to speak, that when not all odd numbers are prime, there can be an infinite number of primes anyway.

I know my brain hurts contemplating that.

To restate the FACTS:

1. The natural number 5 is prime.

2. 5 = 10/2

3. 5*2 = 10

4. Therefore, all numbers ending in 5 or 0 are multiples of 5.

Prime Factors

Though multiplication and its inverse, division, are performed easily with all whole numbers, the principle of equivalency leaves an easier solution to working with many of them.  The practice of "finding the factors" need not end with big numbers. In taking on large numbers, it is perfectly alright to break them up into the numbers from which they came by way of multiplication.

Even Numbers.

One half of all natural numbers are "even," this is to say they can be divided by the number 2.  Two is the first "prime" factor.  Almost everyone remembers the cheer: "Two, four, six, eight, who do we appreciate?"  Well, those are the first four "even" numbers, which hypothetically go on forever.  Along with the the beginning whole number "0," these provide the clue that a number has at least three factors: 1, 2 and the number in question.

Whether you multiply an even or an odd number by an even number, the answer will be even.

For example:  1234 is an even number, and therefor is not a prime number.  Its factors are 1, 2 and 617.  Is 617 a prime number?  Well, it isn't even.  Two is the only "even" prime number, so let us move on to other "prime suspects"

Odd Numbers

The other half of all natural numbers are odd.  This does not mean they are prime, but it makes task of finding big prime numbers a little easier.  I am not one to pursue such a task.  Suffice it to say that an odd number needs to be approached with care.  It can have hidden factors just waiting to be discovered.

Taking the factor of 1234 -- 617 -- the first thing is to find that number's square root.  This is best done with a calculator, but I recognize this as close to 25x25, that is 625.  This sets the limit.  A prime factor would have to be under 25, but not by much.  23x23 has a product 529.

Dropping back to the basics, then, we start with 3.  By theorem, the sum of the digits of any number must be divisible by 3 if the number is divisible by 3. 6+1+7 = 14.  14 is not divisible by 3.

The next prime number is 5. "Counting by fives" is easy, and it reveals to numbers, one even and one odd.  Every even number ending in 0 is divisible by 2 and 5.  This is two for one!  So, 617 is not divisible by 5 either.

The last odd number under 10 is the number 7.  There is no easy way to tell if a number is divisible by 7.  In the case of 617, we see a seven, but the first two numbers return a remainder of 5, leaving 57 (not divisible by seven.  Knowing the multiples of 7 up to at least 9x7 is advisable.

Note that multiplying an odd number by an odd number will get an odd number:

1x1 = 1 5x5 = 25 9x9 = 81
3x3 = 9 5x7 = 35
3x5 =15 5x9 = 45
3x7 =21 7x7 = 49
3x9 =27 7x9 = 63

With the factors 11, 13, 17, 19, and 23, only 11x17 even remotely comes close. However though 11x17 ends in 7, it is far from 617. What about 11x27?  That gets closer, but is far short as well (270 + 27 = 297).

So, 617 is indeed prime.

The most important products to know are those of the prime numbers 2, 3, 5 and 7.  Note, standing by itself is the number 49!  The "new" answer to the universal question!

2|  4

3|  6  9

5| 10 15 25

7| 14 21 35 49

      2  3  5  7 

Finding prime factors:

Starting with 2, what are the prime factors of 7,984,356 (a totally random seven digit number!)

Immediately 2 "works."  Trying 4, we get 1,996,089. Not even, so on to 3.  These digits are 1+9+9+6+0+8+9.  Added this gives us 42; reducing further to 6

So, with factors 2x2x3, we can divide by 12 to get 666,563  Is ths as far as we can go?  Not divisible by 5, so we try 7, 11, 13 and 17.  Seventeen works, yielding 39,139.

This leaves factors of 1, 2, 3, 4, 6, 12, 17, and 39,139.

Using a handy calculator, I know that the square of that large number is just under 198.  197 is a prime number having as its square 38,809. 197 times 199, the next prime number, equals 39,203. This means there are no more prime factors of our chosen number.

The prime factors of 1,996,089 are 2,3,17 and 39,139.


So, what do I know?

Odd x odd = Odd number
Odd x even = Even number
Odd + even = Odd number
Odd + odd = Even number

A prime number is a natural number that has exactly two natural divisors: 1 and itself.

2, 3, 5 and 7 are prime numbers under 10.

Even number 2 is the powerhouse of the primes, affecting ALL even numbers.

Odd number 3 can be seen to be a factor if the sum of the digits add up to a number divisible by 3.

Odd number 5 shouts out from the end of one of its products.  If the even number 0 is there, the 2 and 5 are instantly known.

Seven times seven is forty-nine (7x7=49). Every digit between 1 and 9 shows up as the final digit of multiples of 7.  So, don't look for an easy out here.

All prime numbers larger than 2 are odd numbers. About one in four numbers is prime (in the first 200 natural numbers, at least).

The factors of any number start with one and end with the square root of that number.   

Inverse Behavior

Everything in math has an "opposite."  It is sort of like a religion, with its light and dark, yin and yang, or whatever.  In math, opposites "cancel" each other out.  Sort of like matter and antimatter.  But in doing so, these reactions make math a lot easier.

There are four things one can do with numbers: Add, subtract, multiply and divide.  The latter two are "short cuts" of the former two. That is to say, multiplication is just adding; and division is just subtracting until you come out even or with something left over.

Addition and Subtraction

Opposite Numbers

So, just how do the "inverse" reactions help in math?  First, numbers live in "parallel universes" on either side of the "neutral zone," aka zero.

    -10 -9 -8 -7 -6  -5 -4 -3 -2 -1|    |+1 +2 +3 +4 +5 +6  +7 +8 +9 +10

<-------------------------------------- 0--------------------------------------------->

Addition

The trouble with numbers, is that they only get "stronger" the farther they get from the "neutral zone." This is a great truth, for a number can increase indefinitely by adding just one unit at a time.  The end is not in site, for it is always beyond the biggest number.

But many times, that power is diminished as necessary reversals happen.  If too much momentum is lost, the number changes sides and wears the sign of the opposing "universe"

Inverse reaction (subtraction)

 But for this illustration, numbers are stubborn, they "teleport" into the other universe!  But alas, they are inexplicitly drawn to their counterpart, resulting in annihilation of both of them.

-6+6

=0

Fractions

Trying to get to the other side of the zero by just turning around is not a good path for an signed number either.  As they try to approach zero, they get weaker and weaker. Finally, they reach the "event horizon," (+/-1) and they begin to break into pieces, but never quite die.  Just as they could always advance away from zero, they will be stuck between +/-1 and 0 unless they again turn around and progress away from "absolute" zero.

  < 1/20  2/19  3/18  4/17  5/16  6/15  7/14  8/13  9/12 10/11 

0=======================1

Scotty, now would be a good time.  Beam me up, now!

Back to reality

Alright, enough fun.  Back to reality.  Hopefully, it might help someone "see" how negative numbers fit in to the scheme of things.  You really can add one more forever in each direction.  That is known as infinity.  It is also true that there are an infinite number of divisions between each of the whole numbers and their opposing negative selves.

Going in a negative direction along the line is called Subtraction, the inverse of Addition.  If one were to go over into the opposing "universe" the inverse would be Addition.  When you just lay a negative number next to a positive number, you have a "subtraction" problem using the bigger of the two.  The sign of the bigger number "wins" as the "difference" is determined,'

-8+5 = -(8-5) = -(3) = -3    Notice the subtraction was done inside a negative parenthesis.

+17-13 = +(17-13) = +4     Just for consistency, the same format is used. Positive wins!


I've read that the first functioning "computer" did not add, but subtracted.  It was a mechanical "difference" machine.  That brings us to Division.  Not like a battalion in the war of numbers, but in the process of dividing whole numbers by "natural" numbers.  Only these numbers can be "trusted" for they all follow a fast rule:  To be "rational" a number must be able to exist in the following form

a

b

(where a is an integer and b is a natural number)

What is a natural number?  It is any positive integer.  This excludes 0. which is neutral.  In other words, you cannot divide by 0.  In fact, you cannot divide by anything but a positive integer.

Integers:  {. . . -3, -2, -1, 0, +1,+2, +3 . . .}

Called "signed" numbers, these are all numerals, be they negative or positive.  It includes the "supernatural" number 0  -- just kidding.  As opposed to "natural,"  zero stands apart, but has powers beyond ordinary integers.

Whole Numbers:  {0, 1, 2, 3 ...}

Zero holds it's own as a "whole" number.  It is healthy, and stands in places all other numbers do, but without any voice.  It is barely noticed.  But when missing, the other numbers are greatly diminished.  The zero is second only to the ruling integer, being his "right hand" man.  I know, weak analogy, but alas, it fits.

Natural Numbers:  {1, 2, 3 ...}

One thing zero is forbidden to do is divide.  It is against all the "natural" laws of numbers. It is beyond logic to say that a whole number can be divided into pieces that have NO value.  If  zero were used as a divisor, then math would be impossible.  It is easy to see this.  Consider this:

For x = 0, verify  2x/x = 1.

2x/x = 1

2x*(1/2)= 1*(1/2)

x = 1/2

0  =/= 1/2

2*(x/x) = 1

2*1 = 1

2 =/= 1

Since by definition x/x  is always 1, and 0/x is by definition 0, then 0/0 would become 1!  Something out of nothing?  Nope, not happening.

Division cannot be by a fraction either. In the case of "division by a fraction" the inverse of division is used.  That is, the "fraction" in the denominator is turned over to become its own "reciprocal" and then the inverse of division, that is, multiplication, takes over.

So, what do I know?

+A-A = -A+A = 0

A-A = 0

A/A = 1

A = A/1 (a rational number)

1/A * A/1 = 1

A/B * B/A = 1 

Next: In Their Prime: Factors that matter

Strange Facts about Exponents

Sometimes things just have to be proven.

We are taught to accept some things as fact, but they are not as evident as the facts I know intuitively.  These are the "math facts" that must be derived from the basic things.

This can be done using the "shorthand" known as the exponent.  It is like a "component" but set over to the side.  It is not part of the number but is a reminder that the number has been acted upon by multiplying it by itself.  The common exponents are the "square" [2], and the "cube" [3], drawn from the construction industry: flat panel and a "box."

When working with "whole" numbers, that is to say, zero and all the numbers we count with (aka "natural numbers), the exponent tell us at a glance that things just got a lot bigger.

Exponents also work with what are called "rational" numbers and even with 'irrational" numbers.  And lo, and behold, exponents work with "imaginary numbers" -- all these are things I know, but they need to be shown to be true.

Squares, Cubes and Beyond

First an illustration:

Consider that you are going to build a box, not quite as big as Noah's box (ark) but perhaps about the size of what that ancient Jewish guy, Bezaleel, did with some acacia wood (the box: the ark).  We'll skip the gold.  Anyway, the point is, measurements are made for the box that included its "footprint" (length and width) and its "body" (adding height).  The ends of the box were square, having the same height and width.

Given a width, call it W, then the board will be cut that wide in two (2) directions to get a square.  Or, in this case, W squared.

W x W = W^2

Well unlike Bezaleel or Noah, let us say the instructions to make the box the same measurements in all three dimensions -- put on your 3D glasses here.

Then, keep the measuring using the same mark on the stick, and you construct a cube!

W*W*W = W^3

No, not that "WWW" -- just a cube.  You know, like the description of the Holiest of Holies.  You don't know about that?  (leave a comment, I'll explain).

If you divide a cube by a square, you're back to a line the length of one of the edges.  So how does that help anybody?

Consider this algebraic express:

w^3/w^2 = W*W*W/W*W.

Going a step further:

W/W * W/W * W = 1 * 1 * W = W

OR

W^2/W^2 * W = W

Now, notice the relationship between the cube and the square:

W^3  = W^1.  W^1 = W^[3-2] or simply W.
W^2

The principle is, when dividing exponents, subtract those in the denominator from those in the numerator.

So, when when using the same base, the form N^D/N^D = N^[D-D] = N^0.  But wait, there is that zero again. N^0 = 1.  No matter what N represents!

All that to get to the first know fact of the day:

N^0 = 1.

Then, what happens if the exponent is larger on the bottom?

N^3   = N^[3-5] = N^[-2].
N^5

What in the world?

Simple, really. Look at it the "long" way:

      1*N*N*N     = 1/N*N.  The N^2 is in the denominator.
N*N*N*N*N*1

Think of it using the number 2.

2^3 = 8
2^2 = 4
2^1 = 2
2^0 = 1
2^[-1] = 1/2
2^[-2] = 1/4
2^[-3] = 1/8


So, what do I know?


N*D = multiplying N times D
But
N^D = multiplying by N, D times

N^0 = 1

N^[-D] =     1   
                 N^D

Coming Next: Making Arrangements