(See Primes and Bracelets first)

Based on the characteristics of their bracelets, (the collective behavior of their limited set of repetitive modulo additions), I discovered that there are four distinct groups of prime numbers:

- The first group contains a single prime with the self-complementary {0,1,… } bracelet, the only even prime with the unique self-complementary bracelet of the form {0,n/2,n/2,0}:
- 2: {0,1,

……1,0} of length 3 = n+1 as well as {0,0}

- 2: {0,1,
- The second group consists of another single prime with the exceptionally long self-complementary {0,1,…} bracelet of length 20 (length = 4*n) plus an anomalous 4-member third bracelet, making 5 unique among all prime numbers (and it has nothing to do with our decimal number base as these patterns do not rely on how we represent the integers but are intrinsic to the integers themselves:
- 5: {0,1,1,2,3,0,3,3,1,4,

……0,4,4,3,2,0,2,2,4,1,0}

and {1,3,4,2,1} as well as {0,0}

- 5: {0,1,1,2,3,0,3,3,1,4,
- The third group of primes consists of prime numbers whose decimal representation ends in the digit 3 or the digit 7. They consistently have an {0,1,…} self-complementary bracelet of the length 2(n+1)/x where n is the prime and x is a small integer, usually 1:
- 3: {0,1,1,2,

……0,2,2,1,0} len=8=2*(n+1)

- 7: {0,1,1,2,3,5,1,6,

……0,6,6,5,4,2,6,1,0} len=16=2*(n+1) - 13: {0,1,1,2,3,5,8,0,8,8,3,11,1,12,

……..0,12,12,11,10,8,5,0,5,5,10,2,12,1,0} (len=28=2*(n+1) - 17: {0,1,1,2,3,5,8,13,4,0,4,4,8,12,3,15,1,16,

……..0,16,16,15,14,12,9,4,13,0,13,13,9,5,14,2,16,1,0} len=36=2*(n+1) - 23: {0,1,1,2,3,5,8,13,21,11,9,20,6,3,9,12,21,10,8,18,3,21,1,22,

……..0,22,22,21,20,18,15,10,2,12,14,3,17,20,14,11,2,13,15,5,20,

……………………………………………………………..2,22,1,0} len=48=2*(n+1) - 37: length = 76 = 2*(n+1)
- 43: length = 88 = 2*(n+1)
- 47: length = 32 = 2*(n+1)/3
- 53: length = 108 = 2*(n+1)
- 67: length = 136 = 2*(n+1)
- 73: length = 148 = 2*(n+1)
- 83: length = 168 = 2*(n+1)
- 97: length = 196 = 2*(n+1)
- 103: length = 208 = 2*(n+1)
- 107: length = 72 = 2*(n+1)/3
- 113: length = 76 = 2*(n+1)/3
- 127: length = 256 = 2*(n+1)

- 3: {0,1,1,2,
- The fourth and last group of primes consists of prime numbers whose decimal representation ends in the digit 1 or the digit 9. They consistently have an {0,1,…} bracelet of the length (n-1)/x where n is the prime and x is a small integer, usually 1:
- 11: {0,1,1,2,3,5,8,2,10,1,0} len=10=(n-1)

- 19: {0,1,1,2,3,5,8,13,2,15,17,13,11,5,16,2,18,1,0} len=18=(n-1)
- 29: {0,1,1,2,3,5,8,13,21,5,26,2,28,1,0} len=14=(n-1)/2
- 31: {0,1,1,2,3,5,8,13,21,3,24,27,20,16,5,21,

……..26,16,11,27,7,3,10,13,23,5,28,2,30,1,0} len=30=(n-1) - 41: length = 40 = (n-1) {self-complementary}
- 59: length = 58 = (n-1) {NOT self-complementary}
- 61: length = 60 = (n-1) {self-complementary}
- 71: length = 70 = (n-1) {NOT self-complementary}
- 79: length = 78 = (n-1) {NOT self-complementary}
- 89: length = 44 = (n-1)/2 {self-complementary}
- 101: length = 50 = (n-1)/2 {NOT self-complementary}
- 109: length = 108 = (n-1) {self-complementary}

Since there is only one prime whose last decimal digit is 2, one prime whose last decimal digit is 5, and no primes ending in 0,4,6,or 8, this accounts for all prime numbers.

I’ve checked these prime groupings into the tens of thousands with no primes not conforming to these patterns. However, in checking non-primes, there are extremely rare non-primes that seem to mimic these characteristics. As we venture into the realm of larger integers, it seems to get increasingly more unusual for a non-prime to have a characteristic that might look like that of a prime.

I ran into a few non-primes that might have looked like primes, but, the work being several years ago, I wasn’t sure, so I went back today. The first false prime, using the {0,1,…} bracelet’s characteristics as a test was 377=13*29, but it’s {0,1,…} bracelet was only 28 beads long and **not** self-complementary, as all bracelets for primes ending in 3 or 7 are, but it did fit 27 bracelets 28 beads long into 2*(n+1). As I remember, the first really convincing false prime was in the 600s. This may take a while. Got past 1000 with no even possible false primes but 377 and, as I said earlier, it’s not really a serious fake. Still, I’m reluctant to conclude that this is a way to determine if a number is prime. It weeds out just about all non-primes for sure, but, until this is tested on far more values, I wouldn’t completely rely on it.

I’m fairly certain that, except for 5, all the primes I looked at were fairly regular and followed the expected pattern called for by the trailing digit of its decimal representation.

In any case, I thought that regular and significant behavioral differences between groups of integers was an interesting topic and might be significant to people doing encryption/decryption algorithms, pseudo-random number generators, or looking for different types of humongous primes.

© David N. Dodson, Phoenix, AZ, August 2019