For every n ≥ 2 and every k ≥ 1 , the tabular product satisfies Π T ( k , n ) = n k k ! , hence n | Π T ( k , n ) . The condition

Π T ( k , n ) ≡ 0     ( mod n )

therefore holds for all n and carries no information. The decision rests entirely on the second condition, gcd ( n , k ! ) = 1 , equivalently gcd ( n , k # ) = 1 . Theorems 3 and 4 should be restated with the first condition removed.

Proposition E1 (Validity threshold) Let n ≥ 2 , let K ≥ 1 be an integer and K # = ∏ p ≤ K p the primorial (the product of the primes ≤ K ). The implication

gcd ( n , K # ) = 1     ⇒     n   prime

holds whenever K ≥ n .

Proof. Sufficiency. If K ≥ n and n is composite, its least prime factor p satisfies p ≤ n ≤ K ; thus p | K # and p | n , so gcd ( n , K # ) ≥ p > 1 .

Contrapositively, gcd = 1 forces n prime.

Sharpness. The critical case is the square of a prime. For n = p 2 the least prime factor is p = n . If K < n , no prime ≤ K divides n , so gcd ( n , K # ) = 1 while n is composite: the implication fails.

The adaptive depth is k ( n ) = ⌊ C ( n ) log 2 n ⌋ with C ( n ) = 3 + ⌊ 1 3 log 10 n ⌋ . Since C ( n ) is itself unbounded, growing like 1 3 log 10 n , one has k ( n ) = Θ ( ( log n ) 2 ) , which is still far below n for large; the logarithmic version is therefore necessarily a sieve, not a test. Two explicit instances, both within the range the article claimed reliable:

l n = 101 2 = 10201 : C ( n ) = 4 , k ( n ) = ⌊ 4 log 2 10201 ⌋ = 53 . As 101 > 53 , gcd ( 10201 , 53 # ) = 1 , so the procedure returns prime for a perfect square.

l n = 1000003 × 1000033 = 1000036000099 : C ( n ) = 7 , k ( n ) = 279 . Both prime factors exceed 279, so gcd ( n , 279 # ) = 1 and the procedure returns prime for a composite.

Example 5 (Section 9.2.3 of the original article) presents n = 999999000001 as composite, equal to ( 10 6 + 1 ) ( 10 6 − 1 ) with factor 101 × 9901001 . However

( 10 6 + 1 ) ( 10 6 − 1 ) = 10 12 − 1 = 999999999999 ,

which factors as 3 3 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ 37 ⋅ 101 ⋅ 9901 , while 101 × 9901001 = 1000001101 ; neither equals 999999000001, which is in fact prime. [1] The example should be replaced by a composite with a small prime factor, for instance n = 10 12 − 1 above, whose factor 3 is detected at once.

Theorem E1 (Primality criterion by coprimality; replaces Theorems 3-4) For

n ≥ 2 , let k = ⌊ n ⌋ . Then

n   prime     ⇔     gcd ( n , k # ) = 1.

Proof. By the argument of Proposition E1, with K = ⌊ n ⌋ : the least prime factor p of a composite n satisfies p ≤ n , hence p ≤ ⌊ n ⌋ = k since p is an integer. (Here the depth is the fixed value k = ⌊ n ⌋ , in contrast with the adaptive depth k ( n ) of Section 5.)

Remark E1 (Status of the criterion) Under the bound k = ⌊ n ⌋ the criterion is decision-equivalent to trial division up to n : it decides the same thing on every input, namely the existence of a prime factor ≤ n . It differs as a procedure—a reformulation of primality as coprimality, n prime ⇔ n coprime to the product of the primes ≤ n : by the prime number theorem [1] [2] , log ( k # ) = ϑ ( k ) ~ k , so k # has on the order of n bits. The contribution of this work is the proposed tabular framework and its associated reformulation.

Definition E1 (Logarithmic primorial sieve) For n ≥ 10 , let C ( n ) = 3 + ⌊ 1 3 log 10 n ⌋ , depth k ( n ) = ⌊ C ( n ) log 2 n ⌋ and k ( n ) # = ∏ p ≤ k ( n ) p . The sieve returns composite (certain) when gcd ( n , k ( n ) # ) > 1 ; otherwise, when gcd ( n , k ( n ) # ) = 1 , it reports no prime factor ≤ k ( n ) —inconclusive as to primality. In particular gcd ( n , k ( n ) # ) = 1 does not imply that n is prime. The restriction n ≥ 10 is necessary: for n ≤ 9 the adaptive depth satisfies k ( n ) ≥ n , so the candidate n itself lies among the primes ≤ k ( n ) ; a prime n then divides k ( n ) # and is wrongly reported composite (this affects precisely the primes 2 , 3 , 5 , 7 ). These small cases are settled directly by the exact bound k = ⌊ n ⌋ of Theorem E1, for which k < n , so no prime can be its own factor.

Theorem E2 (Performance, via Mertens) By Mertens’ third theorem [3] , the density of integers with no prime factor ≤ y is

∏ p ≤ y ( 1 − 1 p ) = e − γ ln y ( 1 + o ( 1 ) ) ,

with γ ≈ 0.5772 . Here C = C ( n ) = 3 + ⌊ 1 3 log 10 n ⌋ ~ ln n / ( 3 ln 10 ) , so

y = k ( n ) ∼ C ( n ) ln n ln 2 ~ ( ln n ) 2 3 ln 2 ln 10 ,       ln k ( n ) ~ 2 ln ln n .

The surviving fraction is therefore

S ( n ) ≈ e − γ ln k ( n ) ~ e − γ 2 ln ln n → 0     as   n → ∞ .

Hence the eliminated fraction of composites tends to 1, while the residual S ( n ) stays strictly positive for every finite n and decays like 1 / ln ln n . These two facts are inseparable: they make the procedure an asymptotically near-perfect filter and, at the same time, forbid it the status of a complete test. Numerical validation (uniform sampling per band) confirms the analysis:

At the smallest band the depth already exceeds n ( k ( 10 3 ) = 39 > 31.6 ≈ 10 3 ), so there the sieve is locally complete and eliminates every composite; the asymptotic estimate e − γ / ln k ( n ) , valid for large n , correspondingly overestimates the survival rate in this regime.

Blind set. The composites missed at depth k ( n ) form

B ( n ) = { m   composite : all   prime   factors   of   m   are > k ( n ) } ,

dominated by the semiprimes m = p q with k ( n ) < p ≤ q and containing the squares p 2 with p > k ( n ) . The sieve is blind precisely on B ( n ) , and its cryptographically relevant core—the balanced semiprimes p q with both factors large—is exactly where factoring itself is hard; the perfect squares and the strongly unbalanced semiprimes it also misses are, by contrast, easy to factor.

Complete two-stage test. Hence the correct, efficient architecture: the logarithmic primorial sieve as a first stage (eliminating ≈94% of composites at negligible cost), followed by a completion test on the survivors—deterministic Miller-Rabin [4] below known bounds, or BPSW [5] . The sieve plays the legitimate role of a front-end filter; completeness comes from the modular stage. The depth k ( n ) is tunable: a larger constant lowers the survivor count as e − γ / ln k ( n ) , at the cost of a primorial of size e ϑ ( k ) .

1) The tabular framework. The identities Σ T ( k , n ) = n k ( k + 1 ) / 2 , Δ T ( a , b , n ) = ( b − a ) n and Π T ( k , n ) = n k k ! are exact and retain their unifying and pedagogical value.

2) Carmichael numbers. The rejection result (Theorem 5 of the original article) uses the bound k = ⌊ n ⌋ + 1 , i.e. the regime k ≥ n , and remains valid: a Carmichael number n is squarefree with at least three prime factors, so its least prime factor satisfies p < n 1 / 3 < n and is detected. This property must be attached to the n bound, not to the logarithmic depth, which rejects only those Carmichael numbers having a prime factor ≤ k ( n ) .