Efficient Sequential Composite Generation and Prime Inference in Base Prime Notation (BPN) - noir

Step 1: Generate Sequences A and B
Generate the sequences 𝐴 = 6𝑘 − 1 and 𝐵 = 6𝑘 + 1 up to a limit. Let’s choose a limit of 𝑘 = 10 for illustration.

For simplicity in this example, we’ll use values of 𝐴 AND 𝐵, rather than a symmetrical range implementation for |𝐴| OR |𝐵| using absolute values I believe is more efficient and elegant.

k step	A (6k-1)	B (6k+1)
0	-1	1
1	5	7
2	11	13
3	17	19
4	23	25
5	29	31
6	35	37
7	41	43
8	47	49
9	53	55
10	59	61

Step 2: Identify Primes Using a Sieve-like Method
Primes in 𝐴 and 𝐵 are those numbers that are not divisible by any smaller prime. For simplicity, assume we have already identified primes up to a certain limit:

Prime Sequence	Numbers	Primes
A	5, 11, 17, 23, 29, 35, 41, 47, 53, 59	5, 11, 17, 23, 29, 41, 47, 53, 59
B	7, 13, 19, 25, 31, 37, 43, 49, 55, 61	7, 13, 19, 31, 37, 43, 61

Step 3: Initialize Min-Heap for Composite Numbers
We start by squaring the primes and inserting them into a min-heap. This table tracks the current state of the heap.

BPN Index Step	Numerical Form	Initial Composite	Heap State
1	5	5 * 5 = 25	25
2	7	7 * 7 = 49	25, 49
3	11	11 * 11 = 121	25, 49, 121
4	13	13 * 13 = 169	25, 49, 121, 169
5	17	17 * 17 = 289	25, 49, 121, 169, 289
6	19	19 * 19 = 361	25, 49, 121, 169, 289, 361
…	…	…	…

Step 4: Generate and Track Composite Numbers
Here we extract the smallest composite, generate new composites, and update the heap.

Extracted Composite	New Composites Generated	Updated Heap State
25 (5*5)	5 * 7 = 35, 5 * 11 = 55	35, 49, 121, 169, 289, 361, 55
35 (5*7)	5 * 13 = 65, 7 * 7 = 49	49, 49, 121, 169, 289, 361, 55, 65
49 (7*7)	7 * 11 = 77, 11 * 11 = 121	49, 55, 121, 169, 289, 361, 65, 77
55 (5*11)	5 * 17 = 85, 11 * 13 = 143	65, 77, 121, 143, 169, 289, 361, 85
…	…	…

Step 5: Infer Prime Numbers
Index values that do not produce matching composite values are inferred as primes.

BPN Index	Absolute Value	Composite	Prime?
0	1	No	No
1	5	No	Yes
2	7	No	Yes
3	11	No	Yes
4	13	No	Yes
5	17	No	Yes
6	19	No	Yes
7	23	No	Yes
8	25	Yes	No
9	29	No	Yes
10	31	No	Yes
11	35	Yes	No
12	37	No	Yes
13	41	No	Yes
14	43	No	Yes
15	47	No	Yes
16	49	Yes	No
17	53	No	Yes
18	55	Yes	No
19	59	No	Yes
20	61	No	Yes

Summary
This table-based method helps to visualize and systematically identify composites within the BPN framework. By using sequences 𝐴 and 𝐵, initializing a heap with prime squares, and tracking generated composites, we can efficiently infer primes based on indices that do not produce composite values.