Manuscript Oai279-A@1 · Oai279-A.tex · lemma · a1abc73f

Lemma: Arithmetic validity of the table

Stage 6: the explicit circuit for the frozen input › A finite arithmetic table · lines 845–870 · proof included or linked

Oai279-A@1#lem:table

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Every displayed $F (m)$ is the complete nondecreasing prime list for $m$ .

Proof. The primes occurring are

2, 3, 5, 7, 11, 13.

The primality of $2, 3$ is immediate. For $5, 7$ , only a divisor $2$ needs checking below their square roots. For $11, 13$ , only $2, 3$ need checking. The respective remainders are nonzero.

The composite rows use the identities

4 = 2^{2}, 6 = 2 \cdot 3, 8 = 2^{3}, 9 = 3^{2},

10 = 2 \cdot 5, 12 = 2^{2} \cdot 3, 14 = 2 \cdot 7, 15 = 3 \cdot 5.

Every list is sorted and its product is its row label. Unique factorization proves completeness, including multiplicities.

Displayed formulas in this passage

unlabeled · line 853
$2, 3, 5, 7, 11, 13.$
unlabeled · line 861
$4 = 2^{2}, 6 = 2 \cdot 3, 8 = 2^{3}, 9 = 3^{2},$
unlabeled · line 864
$10 = 2 \cdot 5, 12 = 2^{2} \cdot 3, 14 = 2 \cdot 7, 15 = 3 \cdot 5.$

Nearby in this section

section: A finite arithmetic table