The Primes Matrix and the Curious Behavior of Prime Numbers

The Primes Matrixand the

Curious Behavior of Prime Numbers

Michael McKee

1. Background of the Analysis

In late Spring of 2008, having nothing better to do, I began the analysis of Prime Numbers in order to find some kind of pattern lying therein. My initial attempt to find a pattern utilized Linear Algebra and the representation of Prime Numbers as matrices consisting of their sums and products. While this proved interesting, it did not lead to the results I had envisioned and so I began a new approach.

I knew that every Prime Number greater than 3 could be represented as one of the forms (6a +1) or (6a – 1), for some integer ‘a’. I decided to see what would happen if I added multiples of 6 to each of the Prime Numbers in a simple matrix (see Figure 1 in Appendix A).

As can be seen from Figure 1, The Primes Matrix has positive multiples of 6 as the x-axis, and the Prime Numbers (including 1 and excluding 2 & 3) as the values for the y-axis. Performing a simple sum of the applicable values at each intersection populates the body of the Primes Matrix. Note that only the resulting Prime Numbers are shown in the Primes Matrix – non Prime values have been left blank (more about the construct of the Primes Matrix in the next section).

As I expected, this resulted in the sort of random noise one comes to expect when working with Prime Numbers. However, I noticed the few values I had populated on the other side of the y-axis looked like something interesting. I quickly completed the other side of the y-axis, which resulted in Figure 2 in Appendix A.

This, to me these patterns looked very interesting. I completed eight of the consecutive pages of the Primes Matrix in an Excel spreadsheet and printed out the results. Some careful cutting and pasting resulted in Figure 3 in Appendix A.

This represented only a single quadrant of the entire Primes Matrix. Although it took a bit of time to complete all 4 quadrants (the result of that effort shown in Figure 4 in Appendix A) I now had the full picture of the Primes Matrix and could begin the task of numerical analysis.

2. The Structure of the Primes Matrix

In this section we will concentrate on the structure of the quadrant of the Primes Matrix shown in Figure 3 in Appendix A. The Primes Matrix is constructed using the following:

Multiplication Addition The Prime Numbers (including the number 1) The Integers (0, 1, 2, 3, …) ei = - 1

The values which make-up the y-axis of the Primes Matrix are defined as follows:

The ordered set of Primes Numbers including 1 and excluding 2 and 3

The values which make-up the x-axis of the Primes Matrix are defined as follows:

The product of Prime Numbers 2 and 3 = 6 - multiplied by -

The Integers (0, 1, 2, 3, …)- multiplied by -

ei = - 1

The Primes Matrix is a simple construct made up of the values specified above and the sum of these values at each intersection.

As can be seen in Figure 4 in Appendix A, the 4 quadrants of the Primes Matrix are mirror images on one another. The mirror image of the quadrant shown in Figure 3 in Appendix A has a positive x-axis and negative Prime Numbers.

As we can see in Figure 2 in Appendix A, this construct results in both positive and negative numbers being generated within the body of the Primes Matrix. We also see both Prime and non-Prime numbers

being generated as well. And so this was what I had constructed as of May 2008 when this analysis was postponed for a few months.

3. The Prime Pair Sets

Later in the summer of 2008 I had the opportunity to return to the Primes Matrix to see what I could learn from my previous efforts. I had to re-acquaint myself with what I had done, but it didn’t take too long to find something interesting.

If you look at each column of the negative prime values (Figure 2 in Appendix A; the values in green) you will notice that you can add the largest negative value to the smallest negative value (in that same column) and the sum will be the column value (on the x-axis). You can then take the next largest prime and the next smallest prime and perform the same function yielding the same results. And the next and the next and so on until all of the prime numbers on that column have been exhausted.

In other words, the Prime Pair sets may be read directly from the Primes Matrix. These pairs in each column form the sets of Prime Pair values for that column. The first 80 Prime Pair sets are shown in Appendix B. Although the Prime Pair Sets are read directly from the Primes Matrix as negative values, we can multiply through by -1 to show the members of each set as positive values.

Premise 1: Each column will contain an even number of Prime Numbers.

Premise 2: These Prime Numbers will be evenly distributed between the two forms (6a + 1) and (6b – 1); ‘a’ and ‘b’ are positive integers.

Premise 3: These Prime Numbers will be paired-up within each column of the Primes Matrix so that one of the values will be of the form (6a + 1) and the other of the form (6b – 1).

Premise 4: The sum of the Prime Numbers, which are paired-up, will equal the x-axis value of that column (from the Premise above this would be 6a + 6b). Thus, Premise 3 would be a

necessary condition if the sum of the paired Prime Numbers were to equal the value of the column defined along the x-axis.

Prime Pair Set Definition:

A Prime Pair Set (Sn where n = 1, 2, 3, …) will contain [at least] one member consisting of a pair of Prime Numbers {(Pa, Pb)} where;

Pa is a Prime number defined as 6a + 1 (a = 0, 1, 2, …)Pb is a Prime number defined as 6b - 1 (b = 0, 1, 2, …)

When, Pa + Pb = 6(a + b) = 6n (where n = 1, 2, 3, …)

Example:

The first Prime Pair Set S1 has a single member consisting of the Prime Numbers 1 and 5:

Set 1 Value = 6 S1 = {(1, 5)}

The second Prime Pair Set S2 has two members consisting of the Prime Numbers 1, 5, 7 and 11:

Set 2 Value = 12S2 = {(1, 11),

(5, 7)}

Note:

The set (Pa, Pb) will be indistinct from the set (Pb, Pa) – for our purposes these sets will map to the same member.

Observations:

A Prime Number has a Prime Pair relationship to another Prime Number only [at most] once as defined within one of the columns of the Primes Matrix.

It’s surprising (to me anyway) that these relations between the Prime Numbers can be visualized in the Primes Matrix.

This would be much less elegant (in fact, quite incomplete) were the number 1 not provided its rightful place as a Prime Number.

Why do the Prime Numbers exhibit this behavior? Each successive Prime Number must be only its specific value or the Primes Matrix would not work.

Why do the Prime Pair Sets show-up when subtracting multiples of 6 from the Prime Numbers, but nothing of any consequence when adding multiples of 6 to the same?

Appendix A

Figure 1

Figure 2

Figure 3

Figure 4

Appendix B: The First 80 Prime Pair Sets

Set 1: Value = 6 (1 member)S1 = {(1, 5)}

Set 2: Value = 12 (2 members)S2 = {(1, 11),

(5, 7)}

Set 3: Value = 18 (3 members)S3 = {(1, 17),

(5, 13),(7, 11)}

Set 4: Value = 24 (4 members)S4 = {(1, 23),

(5, 19),(7, 17),(11, 13)}

Set 5: Value = 30 (4 members)S5 = {(1, 29),

(7, 23),(11, 19),(13, 17)}

Set 6: Value = 36 (4 members)S6 = {(5, 31),

(7, 29),(13, 23),(17, 19)}

Set 7: Value = 42 (5 members)S7 = {(1, 41),

(5, 37),(11, 31),(13, 29),(19, 23)}

Set 8: Value = 48 (6 members)S8 = {(1, 47),

(5, 43),(7, 41),(11, 37),(17, 31),(19, 29)}

Set 9: Value = 54 (6 members)S9 = {(1, 53),

(7, 47),(11, 43),(13, 41),(17, 37),(23, 31)}

Set 10: Value = 60 (7 members)S10 = {(1, 59),

(7, 53), (13, 47), (17, 43), (19, 41), (23, 37), (29, 31)}

Set 11: Value = 66 (6 members)S11 = {(5, 61),

(7, 59), (13, 53), (19, 47), (23, 43), (29, 37)}

Set 12: Value = 72 (7 members)S12 = {(1, 71),

(5, 67), (11, 61), (13, 59), (19, 53), (29, 43), (31, 41)}

Set 13: Value = 78 (7 members)S13 = {(5, 73),

(7, 71), (11, 67), (17, 61), (19, 59), (31, 47), (37, 41)}

Set 14: Value = 84 (9 members)S14 = {(1, 83),

(5, 79), (11, 73), (13, 71), (17, 67), (23, 61), (31, 53), (37, 47), (41, 43)}

Set 15: Value = 90 (10 members)S15 = {(1, 89),

(7, 83), (11, 79), (17, 73), (19, 71), (23, 67), (29, 61), (31, 59), (37, 53), (43, 47)}

Set 16: Value = 96 (7 members)S16 = {(7, 89),

(13, 83), (17, 79), (23, 73), (29, 67), (37, 59), (43, 53)}

Set 17: Value = 102 (9 members)S17 = {(1, 101),

(5, 97), (13, 89), (19, 83), (23, 79), (29, 73), (31, 71), (41, 61), (43, 59)}

Set 18: Value = 108 (9 members)S18 = {(1, 107),

(5, 103), (7, 101), (11, 97),

(19, 89), (29, 79), (37, 71), (41, 67), (47, 61)}

Set 19: Value = 114 (11 members)S19 = {(1, 113),

(5, 109), (7, 107), (11, 103), (13, 101), (17, 97), (31, 83), (41, 73), (43, 71), (47, 67), (53, 61)}

Set 20: Value = 120 (12 members)S20 = {(7, 113),

(11, 109), (13, 107), (17, 103), (19, 101), (23, 97), (31, 89), (37, 83), (41, 79), (47, 73), (53, 67), (59, 61)}

Set 21: Value = 126 (10 members)S21 = {(13, 113),

(17, 109), (19, 107), (23, 103), (29, 97), (37, 89), (43, 83), (47, 79), (53, 73), (59, 67)}

Set 22: Value = 132 (10 members)S22 = {(1, 131),

(5, 127), (19, 113), (23, 109), (29, 103), (31, 101), (43, 89), (53, 79), (59, 73), (61, 71)}

Set 23: Value = 138 (9 members)S23 = {(1, 137),

(7, 131), (11, 127), (29 109), (31, 107), (37, 101), (41, 97), (59, 79), (67, 71)}

Set 24: Value = 144 (11 members)S24 = {(5, 139),

(7, 137), (13, 131), (17 127), (31, 113), (37, 107), (41, 103), (43, 101), (47, 97), (61, 83), (71, 73)}

Set 25: Value = 150 (13 members)S25 = {(1, 149),

(11, 139), (13, 137), (19 131), (23, 127), (37, 113), (41, 109), (43, 107), (47, 103),

(53, 97), (61, 89), (67, 83), (71, 79)}

Set 26: Value = 156 (11 members)S26 = {(5, 151),

(7, 149), (17, 139), (19 137), (29, 127), (43, 113), (47, 109), (53, 103), (59, 97), (67, 89), (73, 83)}

Set 27: Value = 162 (10 members)S27 = {(5, 157),

(11, 151), (13, 149), (23 139), (31, 131), (53, 109), (59, 103), (61, 101), (73, 89), (79, 83)}

Set 28: Value = 168 (14 members)S28 = {(1, 167),

(5, 163), (11, 157), (17, 151), (19, 149), (29, 139), (31, 137), (37, 131), (41, 127), (59, 109), (61, 107), (67, 101), (71, 97), (79, 89)}

Set 29: Value = 174 (12 members)S29 = {(1, 173),

(7, 167), (11, 163), (17, 157), (23, 151), (37, 137), (43, 131), (47, 127), (61, 113), (67, 107), (71, 103), (73, 101)}

Set 30: Value = 180 (15 members)S30 = {(1, 179),

(7, 173), (13, 167), (17, 163), (23, 157), (29, 151), (31, 149), (41, 139), (43, 137), (53, 127), (67, 113), (71, 109), (73, 107), (79, 101), (83, 97)}

Set 31: Value = 186 (13 members)S31 = {(5, 181),

(7, 179), (13, 173), (19, 167), (23, 163), (29, 157), (37, 149), (47, 139), (59, 127), (73, 113), (79, 107), (83, 103), (89, 97)}

Set 32: Value = 192 (12 members)S32 = {(1, 191),

(11, 181), (13, 179), (19, 173), (29, 163), (41, 151), (43, 149), (53, 139), (61, 131), (79, 113), (83, 109), (89, 103)}

Set 33: Value = 198 (14 members)S33 = {(1, 197),

(5, 193), (7, 191), (17, 181), (19, 179), (31, 167), (41, 157), (47, 151), (59, 139), (61, 137), (67, 131), (71, 127), (89, 109), (97, 101)}

Set 34: Value = 204 (14 members)S34 = {(5, 199),

(7, 197), (11, 193), (13, 191), (23, 181), (31, 173), (37, 167), (41, 163), (47, 157), (53, 151), (67, 137), (73, 131), (97, 107), (101, 103)}

Set 35: Value = 210 (19 members)S35 = {(11, 199),

(13, 197), (17, 193), (19, 191), (29, 181), (31, 179), (37, 173), (43, 167), (47, 163), (53, 157), (59, 151), (61, 149), (71, 139), (73, 137), (79, 131), (83, 127), (97, 113), (101, 109), (103, 107)}

Set 36: Value = 216 (13 members)S36 = {(5, 211),

(17, 199), (19, 197), (23, 193), (37, 179), (43, 173), (53, 163), (59, 157), (67, 149), (79, 137), (89, 127), (103, 113), (107, 109)}

Set 37: Value = 222 (11 members)S37 = {(11, 211),

(23, 199), (29, 193), (31, 191), (41, 181), (43, 179), (59, 163), (71, 151), (73, 149),

(83, 139), (109, 113)}

Set 38: Value = 228 (13 members)S38 = {(1, 227),

(5, 223), (17, 211), (29, 199), (31, 197), (37, 191), (47, 181), (61, 167), (71, 157), (79, 149), (89, 139), (97, 131), (101, 127)}

Set 39: Value = 234 (16 members)S39 = {(1, 233),

(5, 229), (7, 227), (11, 223), (23, 211), (37, 197), (41, 193), (43, 191), (53, 181), (61, 173), (67, 167), (71, 163), (83, 151), (97, 137), (103, 131), (107, 127)}

Set 40: Value = 240 (19 members)S40 = {(1, 239),

(7, 233), (11, 229), (13, 227), (17, 223), (29, 211), (41, 199), (43, 197), (47, 193),

(59, 181), (61, 179), (67, 173), (83, 157), (89, 151), (101, 139), (103, 137), (109, 131), (113, 127)}

Set 41: Value = 246 (16 members)S41 = {(5, 241),

(7, 239), (13, 233), (17, 229), (19, 227), (23, 223), (47, 199), (53, 193), (67, 179), (73, 173), (79, 167), (83, 163), (89, 157), (97, 149), (107, 139), (109, 137)}

Set 42: Value = 252 (17 members)S42 = {(1, 251),

(11, 241), (13, 239), (19, 233), (23, 229), (29, 223), (41, 211), (53, 199), (59, 193), (61, 191), (71, 181), (73, 179), (79, 173), (89, 163), (101, 151), (103, 149), (113, 139)}

Set 43: Value = 258 (15 members)S43 = {(1, 257),

(7, 251), (17, 241), (19, 239), (29, 229), (31, 227), (47, 211), (59, 199), (61, 197), (67, 191), (79, 179), (101, 157), (107, 151), (109, 149), (127, 131)}

Set 44: Value = 264 (17 members)S44 = {(1, 263),

(7, 257), (13, 251), (23, 241), (31, 233), (37, 227), (41, 223), (53, 211), (67, 197), (71, 193), (73, 191), (83, 181), (97, 167), (101, 163), (107, 157), (113, 151), (127, 137)}

Set 45: Value = 270 (20 members)S45 = {(1, 269),

(7, 263), (13, 257), (19, 251), (29, 241), (31, 239), (37, 233), (41, 229),

(43, 227), (47, 223), (59, 211), (71, 199), (73, 197), (79, 191), (89, 181), (97, 173), (103, 167), (107, 163), (113, 157), (131, 139)}

Set 46: Value = 276 (18 members)S46 = {(5, 271),

(7, 269), (13, 263), (19, 257), (29, 241), (37, 239), (43, 233), (47, 229), (53, 223), (79, 197), (83, 193), (97, 179), (103, 173), (109, 167), (113, 163), (127, 149), (103, 167), (137, 139)}

Set 47: Value = 282 (17 members)S47 = {(1, 281),

(5, 277), (11, 271), (13, 269), (19, 263), (31, 251), (41, 241), (43, 239), (53, 229), (59, 223), (71, 211), (83, 199),

(89, 193), (101, 181), (103, 179), (109, 173), (131, 151)}

Set 48: Value = 288 (17 members)S48 = {(5, 283),

(7, 281), (11, 277), (17, 271), (19, 269), (31, 257), (37, 251), (47, 241), (59, 229), (61, 227), (89, 199), (97, 191), (107, 181), (109, 179), (131, 157), (137, 151), (139, 149)}

Set 49: Value = 294 (20 members)S49 = {(1, 293),

(11, 283), (13, 281), (17, 277), (23, 271), (31, 263), (37, 257), (43, 251), (53, 241), (61, 233), (67, 227), (71, 223), (83, 211), (97, 197), (101, 193), (103, 191), (113, 181), (127, 167), (131, 163), (137, 157)}

Set 50: Value = 300 (21 members)S50 = {(7, 293),

(17, 283), (19, 281), (23, 277), (29, 271), (31, 269), (37, 263), (43, 257), (59, 241), (61, 239), (67, 233), (71, 229), (73, 227), (89, 211), (101, 199), (103, 197), (107, 193), (109, 191), (127, 173), (137, 163), (149, 151)}

Set 51: Value = 306 (15 members)S51 = {(13, 293),

(23, 283), (29, 277), (37, 269), (43, 263), (67, 239), (73, 233), (79, 227), (83, 223), (107, 199), (109, 197), (113, 193), (127, 179), (139, 167), (149, 157)}

Set 52: Value = 312 (18 members)S52 = {(1, 311),

(5, 307), (19, 293), (29, 283),

(31, 281), (41, 271), (43, 269), (61, 251), (71, 241), (73, 239), (79, 233), (83, 229), (89, 223), (101, 211), (113, 199), (131, 181), (139, 173), (149, 163)}

Set 53: Value = 318 (16 members)S53 = {(1, 317),

(5, 313), (7, 311), (11, 307), (37, 281), (41, 277), (47, 271), (61, 257), (67, 251), (79, 239), (89, 229), (107, 211), (127, 191), (137, 181), (139, 179), (151, 167)}

Set 54: Value = 324 (20 members)S54 = {(7, 317),

(11, 313), (13, 311), (17, 307), (31, 293), (41, 283), (43, 281), (47, 277), (53, 271), (61, 263), (67, 257), (73, 251),

(83, 241), (97, 227), (101, 223), (113, 211), (127, 197), (131, 193), (151, 173), (157, 167)}

Set 55: Value = 330 (24 members)S55 = {(13, 317),

(17, 313), (19, 311), (23, 307), (37, 293), (47, 283), (53, 277), (59, 271), (61, 269), (67, 263), (73, 257), (79, 251), (89, 241), (97, 233), (101, 229), (103, 227), (107, 223), (131, 199), (137, 193), (139, 191), (149, 181), (151, 179), (157, 173), (163, 167)}

Set 56: Value = 336 (19 members)S56 = {(5, 331),

(19, 317), (23, 313), (29, 307), (43, 293), (53, 283), (59, 277), (67, 269), (73, 263), (79, 257),

(97, 239), (103, 233), (107, 229), (109, 227), (113, 223), (137, 199), (139, 197), (157, 179), (163, 173)}

Set 57: Value = 342 (17 members)S57 = {(5, 337),

(11, 331), (29, 313), (31, 311), (59, 283), (61, 281), (71, 271), (73, 269), (79, 263), (101, 241), (103, 239), (109, 233), (113, 229), (131, 211), (149, 193), (151, 191), (163, 179)}

Set 58: Value = 348 (17 members)S58 = {(1, 347),

(11, 337), (17, 331), (31, 317), (37, 311), (41, 307), (67, 281), (71, 277), (79, 269), (97, 251), (107, 241), (109, 239), (137, 211), (149, 199), (151, 197), (157, 191),

(167, 181)}

Set 59: Value = 354 (21 members)S59 = {(1, 353),

(5, 349), (7, 347), (17, 337), (23, 331), (37, 317), (41, 313), (43, 311), (47, 307), (61, 293), (71, 283), (73, 281), (83, 271), (97, 257), (103, 251), (113, 241), (127, 227), (131, 223), (157, 197), (163, 191), (173, 181)}

Set 60: Value = 360 (23 members)S60 = {(1, 359),

(7, 353), (11, 349), (13, 347), (23, 337), (29, 331), (43, 317), (47, 313), (53, 307), (67, 293), (79, 281), (83, 277), (89, 271), (97, 263), (103, 257), (109, 251), (127, 233), (131, 229), (137, 223), (149, 211),

(163, 197), (167, 193), (179, 181)}

Set 61: Value = 366 (18 members)S61 = {(7, 359),

(13, 353), (17, 349), (19, 347), (29, 337), (53, 313), (59, 307), (73, 293), (83, 283), (89, 277), (97, 269), (103, 263), (109, 257), (127, 239), (137, 229), (139, 227), (167, 199), (173, 193)}

Set 62: Value = 372 (18 members)S62 = {(5, 367),

(13, 359), (19, 353), (23, 349), (41, 331), (59, 313), (61, 311), (79, 293), (89, 283), (101, 271), (103, 269), (109, 263), (131, 241), (139, 233), (149, 223), (173, 199), (179, 193), (181, 191)}

Set 63: Value = 378 (22 members)S63 = {(5, 373),

(11, 367), (19, 359), (29, 349), (31, 347), (41, 337), (47, 331), (61, 317), (67, 311), (71, 307), (97, 281), (101, 277), (107, 271), (109, 269), (127, 251), (137, 241), (139, 239), (149, 229), (151, 227), (167, 211), (179, 199), (181, 197)}

Set 64: Value = 384 (20 members)S64 = {(1, 383),

(5, 379), (11, 373), (17, 367), (31, 353), (37, 347), (47, 337), (53, 331), (67, 317), (71, 313), (73, 311), (101, 283), (103, 281), (107, 277), (113, 271), (127, 257), (151, 233), (157, 227), (173, 211), (191, 193)}

Set 65: Value = 390 (28 members)S65 = {(1, 389),

(7, 383), (11, 379), (17, 373), (23, 367), (31, 359), (37, 353), (41, 349), (43, 347), (53, 337), (59, 331), (73, 317), (79, 311), (83, 307), (97, 293), (107, 283), (109, 281), (113, 277), (127, 263), (139, 251), (149, 241), (151, 239), (157, 233), (163, 227), (167, 223), (179, 211), (191, 199), (193, 197)}

Set 66: Value = 396 (21 members)S66 = {(7, 389),

(13, 383), (17, 379), (23, 373), (29, 367), (37, 359), (43, 353), (47, 349), (59, 337), (79, 317), (83, 313), (89, 307), (103, 293), (113, 283), (127, 269), (139, 257), (157, 239),

(163, 233), (167, 229), (173, 223), (197, 199)}

Set 67: Value = 402 (18 members)S67 = {(1, 401),

(5, 397), (13, 389), (19, 383), (23, 379), (29, 373), (43, 359), (53, 349), (71, 331), (89, 313), (109, 293), (131, 271), (139, 263), (151, 251), (163, 239), (173, 229), (179, 223), (191, 211)}

Set 68: Value = 408 (20 members)S68 = {(7, 401),

(11, 397), (19, 389), (29, 379), (41, 367), (59, 349), (61, 347), (71, 337), (97, 311), (101, 307), (127, 281), (131, 277), (137, 271), (139, 269), (151, 257), (157, 251), (167, 241), (179, 229), (181, 227), (197, 211)}

Set 69: Value = 414 (21 members)S69 = {(5, 409),

(13, 401), (17, 397), (31, 383), (41, 373), (47, 367), (61, 353), (67, 347), (83, 331), (97, 317), (101, 313), (103, 311), (107, 307), (131, 283), (137, 277), (151, 263), (157, 257), (163, 251), (173, 241), (181, 233), (191, 223)}

Set 70: Value = 420 (31 members)S70 = {(1, 419),

(11, 409), (19, 401), (23, 397), (31, 389), (37, 383), (41, 379), (47, 373), (53, 367), (61, 359), (67, 353), (71, 349), (73, 347), (83, 337), (89, 331), (103, 317), (107, 313), (109, 311), (113, 307), (127, 293), (137, 283),

(139, 281), (149, 271), (181, 233), (151, 269), (157, 263), (163, 257), (179, 241), (181, 239), (191, 229), (193, 227), (197, 223)}

Set 71: Value = 426 (21 members)S71 = {(5, 421),

(7, 419), (17, 409), (29, 397), (37, 389), (43, 383), (47, 379), (53, 373), (59, 367), (67, 359), (73, 353), (79, 347), (89, 337), (109, 317), (113, 313), (149, 277), (157, 269), (163, 263), (193, 233), (197, 229), (199, 227)}

Set 72: Value = 432 (20 members)S72 = {(1, 431),

(11, 421), (13, 419), (23, 409), (31, 401), (43, 389), (53, 379), (59, 373), (73, 359), (79, 353),

(83, 349), (101, 331), (139, 293), (149, 283), (151, 281), (163, 269), (181, 251), (191, 241), (193, 239), (199, 233)}

Set 73: Value = 438 (21 members)S73 = {(5, 433),

(7, 431), (17, 421), (19, 419), (29, 409), (37, 401), (41, 397), (59, 379), (71, 367), (79, 359), (89, 349), (101, 337), (107, 331), (127, 311), (131, 307), (157, 281), (167, 271), (181, 257), (197, 241), (199, 239), (211, 227)}

Set 74: Value = 444 (22 members)S74 = {(1, 443),

(5, 439), (11, 433), (13, 431), (23, 421), (43, 401), (47, 397), (61, 383), (71, 373), (97, 347), (107, 337),

(113, 331), (127, 317), (131, 313), (137, 307), (151, 293), (163, 281), (167, 277), (173, 271), (181, 263), (193, 251), (211, 233)}

Set 75: Value = 450 (28 members)S75 = {(1, 449),

(7, 443), (11, 439), (17, 433), (19, 431), (29, 421), (31, 419), (41, 409), (53, 397), (61, 389), (67, 383), (71, 379), (83, 367), (97, 353), (101, 349), (103, 347), (113, 337), (137, 313), (139, 311), (157, 293), (167, 283), (173, 277), (179, 271), (181, 269), (193, 257), (199, 251), (211, 239), (223, 227)}

Set 76: Value = 456 (24 members)S76 = {(7, 449),

(13, 443), (17, 439),

(23, 433), (37, 419), (47, 409), (59, 397), (67, 389), (73, 383), (83, 373), (89, 367), (97, 359), (103, 353), (107, 349), (109, 347), (139, 317), (149, 307), (163, 293), (173, 283), (179, 277), (193, 263), (199, 257), (223, 233), (227, 229)}

Set 77: Value = 462 (29 members)S77 = {(1, 461),

(5, 457), (13, 449), (19, 443), (23, 439), (29, 433), (31, 431), (41, 421), (43, 419), (53, 409), (61, 401), (73, 389), (79, 383), (83, 379), (89, 373), (103, 359), (109, 353), (113, 349), (131, 331), (149, 313), (151, 311), (179, 283), (181, 281),

(191, 271), (193, 269), (199, 263), (211, 251), (223, 239), (229, 233)}

Set 78: Value = 468 (25 members)S78 = {(1, 467),

(5, 463), (7, 461), (11, 457), (19, 449), (29, 439), (37, 431), (47, 421), (59, 409), (67, 401), (71, 397), (79, 389), (89, 379), (101, 367), (109, 359), (131, 337), (137, 331), (151, 317), (157, 311), (191, 277), (197, 271), (199, 269), (211, 257), (227, 241), (229, 239)}

Set 79: Value = 474 (22 members)S79 = {(7, 467),

(11, 463), (13, 461), (17, 457), (31, 443), (41, 433), (43, 431), (53, 421), (73, 401), (101, 373), (107, 367),

(127, 347), (137, 337), (157, 317), (163, 311), (167, 307), (181, 293), (191, 283), (193, 281), (197, 277), (211, 263), (223, 251)}

Set 80: Value = 480 (30 members)S80 = {(1, 479),

(13, 467), (17, 463), (19, 461), (23, 457), (31, 449), (37, 443), (41, 439), (47, 433), (59, 421), (61, 419), (71, 409), (79, 401), (83, 397), (97, 383), (101, 379), (107, 373), (113, 367), (127, 353), (131, 349), (149, 331), (163, 317), (167, 313), (173, 307), (197, 283), (199, 281), (211, 269), (223, 257), (229, 251), (239, 241)}