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 (2010-01-19 10:47:32)