Editing Flow-Diagrams (section)

== 3. Prime Generation — Sieve Loop ==

'''Program:''' <code>primesgen.cbl</code> (r80-series paragraphs) + <code>primes.cbl</code> (r81, r83)<br />
'''What it shows:''' The trial-division sieve — how each candidate is tested, how composites are skipped, and how confirmed primes are persisted.

'''Algorithm in words:''' 1. Start at candidate = 3. Load first divisor (prime at ident 1 = prime 2, assumed pre-seeded). 2. Divide candidate by current divisor. 3. If remainder = 0 → composite. Add 2, reset divisor to first prime. 4. If divisor &gt; √candidate → prime confirmed. Write to DB and print buffer. Add 2, reset divisor. 5. Otherwise → try next divisor (fetch next prime from DB by ident).

<pre class="mermaid">flowchart TD
    A([r91-start-primes-generation]) --&gt; B[CALL primes 'connect']
    B --&gt; C{dal-method-ok?}
    C --&gt;|No| FAIL([session-result = 1\nEXIT])
    C --&gt;|Yes| D[test-divider = 2\nold-ident = 1\ntest-number = 3\ntest-number-sqr = SQRT 3]
    D --&gt; LOG1[LOG 'Database initialisation succeeded.']

    LOG1 --&gt; LOOP{test-number\n= 999999999?}
    LOOP --&gt;|Yes| CLOSE([r99-close-primes\nEXIT PROGRAM])

    LOOP --&gt;|No| DIV[r80-test-number\nDIVIDE test-number BY test-divider\nGIVING test-quot REMAINDER test-rest]

    DIV --&gt; EVAL{EVALUATE TRUE}

    EVAL --&gt;|test-rest = 0\ncomposite| NEXT[r82-next-test-number]
    EVAL --&gt;|test-divider &gt; test-number-sqr\nprime confirmed| WRITE[r85-write-prime\nCALL primes 'write'\nINSERT INTO primes prime\nCALL primesui 'write'\nadd to print buffer]
    EVAL --&gt;|otherwise\ntry next divisor| GETDIV[r89-get-next-divider\nCALL primes 'next-divider'\nSELECT prime WHERE ident = old-ident+1]

    WRITE --&gt; NEXT
    GETDIV --&gt; LOOP

    NEXT --&gt; NA[ADD 2 TO test-number\nCOMPUTE test-number-sqr = test-number ** 0.5\nMOVE 1 TO old-ident\nr89-get-next-divider\nload first divisor]
    NA --&gt; LOOP</pre>
'''Data flow through the sieve:'''

<pre>test-number (candidate)     ──→ DIVIDE ──→ test-rest
test-divider (from DB)      ──→ DIVIDE ──→ test-quot
test-number-sqr (SQRT)      ──→ compare to test-divider
old-ident (DB row pointer)  ──→ SELECT prime WHERE ident = old-ident+1 ──→ test-divider
prime-number (confirmed)    ──→ INSERT INTO primes (prime) ──→ database
                            ──→ u-number / u-sequence ──→ primesui print buffer</pre>

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