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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

Objective: Find prime numbers below 100, and the

non-prime or factorable numbers.

I'm going to use an exhaustive approach, that

finds each non-prime number once and only once

without getting double "hits".

You will see the flimsy algorithm unfold below

as utilizing combinations of factors in an

numerical way with the lowest primes being used

first.

Start with 2, and go up.

If multiply to over 100, then move on...

First just list the solely 2-factors: (2) 4 8 16 32 64 128

The next # is 3, and is prime since it is not above in list.

One 3-factor and zero-plus 2-factors: (3) 6 12 24 48 96 192

Two 3-factors and zero-plus 2-factors: 9 18 36 72 144

Three 3-factors and zero-plus 2-factors: 27 54 108

Four 3-factors and zero-plus 2-factors: 81 162

Next #4 is in list above, not prime, it is

4 = 2 times 2.

Next #5 is prime, not listed above.

One 5-factor and no 3-factors and zero-plus 2-factors: (5) 10 20 40 80 160

One 5-factor and one 3-factor and zero-plus 2-factors: 15 30 60 120

One 5-factor and two 3-factors and zero-plus 2-factors: 45 90 180

One 5-factor and three 3-factors and zero-plus 2-factors: 135 270

Two 5-factors and no 3-factors and zero-plus 2-factors: 25 50 100 200

Two 5-factors and one 3-factor and zero-plus 2-factors: 75 150

Two 5-factors and two 3-factors and zero-plus 2-factors: 225 450

Three 5-factors and and no 3-factors and zero-plus 2-factors: 125 250

The next # is 6, and is in a list above, so not prime,

6 = 3 times 2.

Next the #7 is not listed above so it is prime.

One 7-factor and no 5-factors and no 3-factors and zero-plus 2-factors: (7) 14 28 56 112

One 7-factor and no 5-factors and one 3-factor and zero-plus 2-factors: 21 42 84 168

One 7-factor and no 5-factors and two 3-factors and zero-plus 2-factors: 63 126

One 7-factor and no 5-factors and three 3-factors and zero-plus 2-factors: 189 378

One 7-factor and one 5-factor and no 3-factors and zero-plus 2-factors: 35 70 140

One 7-factor and one 5-factor and one 3-factor and zero-plus 2-factors: 105 210

Two 7-factors and no 5-factors and no 3-factors and zero-plus 2-factors: 49 98 196

Two 7-factors and no 5-factors and one 3-factor and zero-plus 2-factors: 147 294

Next # is 8 and it is in the list of 2-factors above, so it is not prime.

Then the # 9 is also listed above, so it is not prime, it is

3 times 3.

The # 10 is listed above, so it is not prime, it is

5 times 2.

Next 11 is prime as it is not listed above.

(Now I will abbreviate "factor" with just "f")

One 11-f and no 7-f and no 5-f and no 3-f and zero-plus 2-f: (11) 22 44 88 176

One 11-f and no 7-f and no 5-f and one 3-f and zero-plus 2-f: 33 66 132

One 11-f and no 7-f and no 5-f and two 3-f and zero-plus 2-f: 99 198

One 11-f and no 7-f and no 5-f and three 3-f and zero-plus 2-f: 297 594

One 11-f and no 7-f and one 5-f and no 3-f and zero-plus 2-f: 55 110

One 11-f and no 7-f and two 5-f and no 3-f and zero-plus 2-f: 275 550

One 11-f and one 7-f and no 5-f and no 3-f and zero-plus 2-f: 77 154

One 11-f and one 7-f and no 5-f and one 3-f and zero-plus 2-f: 231 462

(I am going a bit over 100 here just out of interest and curiosity)

One 11-f and one 7-f and one 5-f and no 3-f and zero-plus 2-f: 385 770

One 11-f and one 7-f and one 5-f and one 3-f and zero-plus 2-f: 1155 2310

Two 11-f and no 7-f and no 5-f and no 3-f and zero-plus 2-f: 121 242

Next is 12, which is listed above, so it is not prime, and is

3 times 2 times 2.

(I will abbreviate "zero-plus" with "zp" hereafter.)

Next # is 13, which is not listed above, so it is a prime number.

One 13-f and no 11-f and no 7-f and no 5-f and no 3-f and zp 2-f: (13) 26 52 104

One 13-f and no 11-f and no 7-f and no 5-f and one 3-f and zp 2-f: 39 78 156

(From hereon out I will not type the "no" quantities; wastes space)

One 13-f and one 5-f and zp 2-f: 65 130

One 13-f and one 7-f and zp 2-f: 91 182

One 13-f and one 11-f and zp 2-f: 143 286

Next # is 14, which is not a prime number as it is listed above.

The #'s 15 and 16 are not prime either as they are listed above.

The #17 is prime as it is not listed above.

One 17-f and zp 2-f: (17) 34 68 136

One 17-f and one 3-f and zp 2-f: 51 102

One 17-f and one 5-f and zp 2-f: 85 170

One 17-f and one 7-f and zp 2-f: 119 238

#18 is above listed.

The # 19 is prime, not above.

One 19-f and zp 2-f: (19) 38 76 152

One 19-f and one 3-f and zp 2-f: 57 114

One 19-f and one 5-f and zp 2-f: 95 190

One 19-f and one 7-f and zp 2-f: 133 266

Numbers 20, 21, and 22 are found above, so they are not prime.

The number 23 is prime and is not listed above.

One 23-f and zp 2-f: (23) 46 92 184

One 23-f and one 3-f and zp 2-f: 69 138

One 23-f and one 5-f and zp 2-f: 115 230

Numbers 24, 25, 26, 27, and 28 are found above, so are factorable and not prime.

The number 29 is the next prime and is not listed above.

One 29-f and zp 2-f: (29) 58 116

One 29-f and one 3-f and zp 2-f: 87 174

One 29-f and one 5-f and zp 2-f: 145 290

The number 30 is not prime as it is

found under the 5's section above.

The number 31 is prime as it is not above.

One 31-f and zp 2-f: (31) 62 124

One 31-f and one 3-f and zp 2-f: 93 186

One 31-f and one 5-f and zp 2-f: 155 310

The numbers 32, 33, 34, 35, and 36 are above so they are not prime.

The number 37 is prime as it is not above.

One 37-f and zp 2-f: (37) 74 148

One 37-f and 3-f and zp 2-f: 111 222

The numbers 38, 39, and 40 are found above, so they are not prime.

The number 41 is prime since it is not above.

One 41-f and zp 2-f: (41) 82 164

One 41-f and 3-f and zp 2-f: 123 246

The # 42 is found in the 7's section above so it is not prime.

The number 43 is not found above, even though 143 is, so 43 is prime.

One 43-f and zp 2-f: (43) 86 172

The #'s 44, 45, and 46 are not prime as they are above.

The # 47 is prime as it is not above.

One 47-f and zp 2-f: (47) 94 188

The numbers 48, 49, 50, 51, and 52 are listed above, so they are factorable,

or non-primes.

All the factorable non-prime numbers below 100 should be listed above

by now, as we are up to fifty, and we have been doubling everything!

So if you want to find a prime number between 50 and 100, just search

for it above, and if it is not found, it is prime.

Let's see if this is true. I'll list the ones not shown above.

I wrote all the numbers listed above that were between 50 and 100,

inclusive, and then found that 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97

were not listed above, so they are prime numbers. The numbers check

with some other internet sources, so that's good.

*Last edited by John E. Franklin (2010-01-18 11:47:32)*

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

I've been reviewing print-outs of this, and the data structure "tree's" comes to mind when

filling out the non-prime tables in a reasonable order. The higher you get up the tree of

primes, then you have all the primes below it and itself for the sons and daughter member

branches for use in remultiplying. I'll work on a better algorithm soon as time permits and

try to explain it. "Tree's" are like structures in computer programming for things such as

hash sorts, etc.

**igloo** **myrtilles** **fourmis**

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

I take it you have not heard of the sieve of Eratosthenes.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

Actually, that's where I got the idea!! But with my way, you have to go to halfway, not the square root, so my

way is more work! And my way doesn't cross off all the even numbers first, just the powers of 2, so there are

differences. But you are right that they are basically the same in that we are crossing out the non-primes! When

I first learned about non-primes, I was very interested that the number of factors was so small.

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

I guess the point of this exercise was to show you could go through

the non-primes using combinations such that you don't come up with

the same non-prime number twice or three times due to it's factors,

as with the Sieve method. But more than that, I am on a personal

mission to increase my awareness on the subject by doing, more than

by reading because I am a really poor reader.

**igloo** **myrtilles** **fourmis**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,750

John E. Franklin wrote:

But with my way, you have to go to halfway, not the square root, so my way is more work!

I alone have the problem of long solutions. When someone comes up with a shorter demonstration than my own, I just scoff.

I am on a personal mission to increase my awareness on the subject by doing, more than by reading because I am a really poor reader.

I like the idea of doing, as I am a poor reader too. I read alot and come away with little.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

**Online**

**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

Thanks for the comments bobby.

I think my next step will be to try to list all non-primes without

knowing the primes up to a 1000 on paper. But this time, I'm

going to do lots of things in parallel, and drop and pick them up

as my count increases. I think I might drop 2's and 3's because

you can tell those at a glance, due to even #'s or adding up the

digits. Just gotta find some really huge paper pad to start my

work on...

**igloo** **myrtilles** **fourmis**

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Just gotta find some really huge paper pad to start my work on...

As part of a fictitious biography:

One of his most important discoveries, still in use today, was that you can turn your paper sideways. This allowed mathematicians to work on even more complex formulas than they were previously capable.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

Good one! Or go diagonally, as the work gets harder and easier, in an oscillatory pattern!

**igloo** **myrtilles** **fourmis**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,750

There was a good scene in "A Beautiful Mind" where JFN efficiently covered an entire blackboard. He even wrote inside the zeros!

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

**Online**

**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

I've worked out the primes up to 500 by skipping products with 2, 3, or 5's in them, and just

starting with 7, 11, 13, 17, etc... Picking up and dropping data for reuse so far is not that hard.

I'll keep going towards 750 or 1000 tomorrow. I'm multiplying each time instead of adding so

I don't get any accumulative errors, just possible singular errors. Data checks with internet though!

It's interesting to me that Daniel Tammet knows all of these up to ten thousand by heart with

colors and feelings, and shapes, and light visualizations.

**igloo** **myrtilles** **fourmis**

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**valona****Member**- Registered: 2010-01-21
- Posts: 1

just go in diagonally,it gets easier

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

I'm up to 661 now. Diagonally? I'll keep that in mind.

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

By using parallel multiplication tables, you can have a lot of

fun looking at the non-prime numbers!!!

It's not a waste of time just because the Sieve has already been done before.

Because the multiplication tables are very important.

Maybe someday we will discover more short-cuts for certain numbers if

we keep trying.

Here's a picture of several parallel multiplication tables centering

around the 500 to 530 mark. 500 yellow marker, 530 red marker.

The numbers in parenthesis are just there so you know what mult

table you are in fast.

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

I have over 12 windows on my screen of columns of numbers now and

I just thought of something special Windows could do to the interface.

A button like the minimize button, except, the window simply goes behind

the other windows on the screen. If this exists, I'd love to know about

it, in any operating system. Anyone know.

Say you have Eight slim windows open like shown above in my

previous post, and then you have eight more on top of them.

Is there a way to make each window go behind as far as it can toward

the iconic desktop?? Even if it takes 8 quick clicks, that is fine, or

hot-keys (accelerators) ?

(The windows would remain "open", just go behind other windows...)

(That way, you could do the same to what came to the front when

the time was appropriate, and move it to the back of the screen)

(...kind of like a juggling act...)

*Last edited by John E. Franklin (2010-01-29 05:36:29)*

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,582

(:

**igloo** **myrtilles** **fourmis**

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