Why primality is polynomial time, but factorisation is not

Differentiating between the signature of a number and its value

A brief review: The significance of evidence-based reasoning

In a paper: The truth assignments that differentiate human reasoning from mechanistic reasoning: The evidence-based argument for Lucas’ Gödelian thesis’, which appeared in the December 2016 issue of Cognitive Systems Research [An16], I briefly addressed the philosophical challenge that arises when an intelligence—whether human or mechanistic—accepts arithmetical propositions as true under an interpretation—either axiomatically or on the basis of subjective self-evidence—without any specified methodology for objectively evidencing such acceptance in the sense of Chetan Murthy and Martin Löb:

“It is by now folklore … that one can view the values of a simple functional language as specifying evidence for propositions in a constructive logic …” … Chetan. R. Murthy: [Mu91], \S 1 Introduction.

“Intuitively we require that for each event-describing sentence, $\phi_{o^{\iota}}n_{\iota}$ say (i.e. the concrete object denoted by $n_{\iota}$ exhibits the property expressed by $\phi_{o^{\iota}}$), there shall be an algorithm (depending on I, i.e. $M^{*}$) to decide the truth or falsity of that sentence.” … Martin H Löb: [Lob59], p.165.

Definition 1 (Evidence-based reasoning in Arithmetic): Evidence-based reasoning accepts arithmetical propositions as true under an interpretation if, and only if, there is some specified methodology for objectively evidencing such acceptance.

The significance of introducing evidence-based reasoning for assigning truth values to the formulas of a first-order Peano Arithmetic, such as PA, under a well-defined interpretation (see Section 3 in [An16]), is that it admits the distinction:

(1) algorithmically verifiable truth’ (Definition 2}); and

(2) algorithmically computable truth’ (Definition 3).

Definition 2 (Deterministic algorithm): A deterministic algorithm computes a mathematical function which has a unique value for any input in its domain, and the algorithm is a process that produces this particular value as output.

Note that a deterministic algorithm can be suitably defined as a realizer‘ in the sense of the Brouwer-Heyting-Kolmogorov rules (see [Ba16], p.5).

For instance, under evidence-based reasoning the formula $[(\forall x)F(x)]$ of the first-order Peano Arithmetic PA must always be interpreted weakly under the classical, standard, interpretation of PA (see [An16], Theorem 5.6) in terms of algorithmic verifiability (see [An16], Definition 1); where, if the PA-formula $[F(x)]$ interprets as an arithmetical relation $F^{*}(x)$ over $N$:

Definition 2 (Algorithmic verifiability): The number-theoretical relation $F^{*}(x)$ is algorithmically verifiable if, and only if, for any natural number $n$, there is a deterministic algorithm $AL_{(F,\ n)}$ which can provide evidence for deciding the truth/falsity of each proposition in the finite sequence $\{F^{*}(1), F^{*}(2), \ldots, F^{*}(n)\}$.

Whereas $[(\forall x)F(x)]$ must always be interpreted strongly under the finitary interpretation of PA (see [An16], Theorem 6.7) in terms of algorithmic computability ([An16], Definition 2), where:

Definition 3 (Algorithmic computability): The number theoretical relation $F^{*}(x)$ is algorithmically computable if, and only if, there is a deterministic algorithm $AL_{F}$ that can provide evidence for deciding the truth/falsity of each proposition in the denumerable sequence $\{F^{*}(1), F^{*}(2), \ldots\}$.

The significance of the distinction between algorithmically computable reasoning based on algorithmically computable truth, and algorithmically verifiable reasoning based on algorithmically verifiable truth, is that it admits the following, hitherto unsuspected, consequences:

(i) PA has two well-defined interpretations over the domain $N$ of the natural numbers (including $0$):

(a) the weak non-finitary standard interpretation $I_{PA(N, SV)}$ ([An16], Theorem 5.6),

and

(b) a strong finitary interpretation $I_{PA(N, SC)}$ ([An16], Theorem 6.7);

(ii) PA is non-finitarily consistent under $I_{PA(N, SV)}$ ([An16], Theorem 5.7);

(iii) PA is finitarily consistent under $I_{PA(N, SC)}$ ([An16], Theorem 6.8).

The significance of evidence-based reasoning for Computational Complexity

In this investigation I now show the relevance of evidence-based reasoning, and of distinguishing between algorithmically verifiable and algorithmically computable number-theoretic functions (as defined above), for Computational Complexity is that it assures us a formal foundation for placing in perspective, and complementing, an uncomfortably counter-intuitive entailment in number theory—Theorem 2 below—which has been treated by conventional wisdom as sufficient for concluding that the prime divisors of an integer cannot be proven to be mutually independent.

However, I show there that such informally perceived barriers are, in this instance, illusory; and that admitting the above distinction illustrates:

(a) Why the prime divisors of an integer are mutually independent Theorem 2;

(b) Why determining whether the signature (Definition 3 below) of a given integer $n$—coded as the key in a modified Bazeries-cylinder (see Definition 7 of this paper) based combination lock—is that of a prime, or not, can be done in polynomial time $O(log_{_{e}}n)$ (Corollary 4 of this paper); as compared to the time $\ddot{O}(log_{_{e}}^{15/2}n)$ given by Agrawal et al in [AKS04], and improved to $\ddot{O}(log_{_{e}}^{6}n)$ by Lenstra and Pomerance in [LP11], for determining whether the value of a given integer $n$ is that of a prime or not.

(c) Why it can be cogently argued that determining a factor of a given integer cannot be polynomial time.

Definition 4 (Signature of a number): The signature of a given integer $n$ is the sequence $a_{_{n,i}}$ where $n + a_{_{n,i}} \equiv 0\ mod\ (p_{_{i}})$ for all primes $p_{_{i}}\ such\ that\ 1\leq i \leq \pi(\sqrt{n})$.

Unique since, if $p_{_{\pi(\sqrt{m})+1}}^{2} > m \geq p_{_{\pi(\sqrt{m})}}^{2}$ and $p_{_{\pi(\sqrt{n})+1}}^{2} > n \geq p_{_{\pi(\sqrt{n})}}^{2}$ have the same signature, then $|m - n| = c_{_{1}}.\prod_{i=1}^{\pi(\sqrt{m})}p_{_{i}} = c_{_{2}}.\prod_{i=1}^{\pi(\sqrt{n})}p_{_{i}}$; whence $c_{_{1}} = c_{_{2}} = 0$ since $\prod_{i=1}^{k}p_{_{i}} > (\prod_{i=2}^{k-2}p_{_{i}}).p_{_{k}}^{^{2}} > p_{_{k+1}}^{2}$ for $k > 4$ by appeal to Bertrand’s Postulate $2.p_{_{k}} > p_{_{k+1}}$; and the uniqueness is easily verified for $k \leq 4$.

Definition 5 (Value of a number): The value of a given integer $n$ is any well-defined interpretation—over the domain of the natural numbers—of the (unique) numeral $[n]$ that represents $n$ in the first-order Peano Arithmetic PA.

We note that Theorem 2 establishes a lower limit for [AKS04] and [LP11], because determining the signature of a given integer $n$ does not require knowledge of the value of the integer as defined by the Fundamental Theorem of Arithmetic.

Theorem 1: (Fundamental Theorem of Arithmetic): Every positive integer $n > 1$ can be represented in exactly one way as a product of prime powers:

$n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}}$

where $p_{1} < p_{2} < \ldots < p_{k}$ are primes and the $n_{i}$ are positive integers (including $0$).

Are the prime divisors of an integer mutually independent?

In this paper I address the query:

Query 1: Are the prime divisors of an integer $n$ mutually independent?

Definition 6 (Independent events): Two events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of the other.

Intuitively, the prime divisors of an integer seem to be mutually independent by virtue of the Fundamental Theorem of Arithmetic

Moreover, the prime divisors of $n$ can also be seen to be mutually independent in the usual, linearly displayed, Sieve of Eratosthenes, where whether an integer $n$ is crossed out as a multiple of a prime $p$ is obviously independent of whether it is also crossed out as a multiple of a prime $q \neq p$:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 …

Despite such compelling evidence, conventional wisdom appears to accept as definitive the counter-intuitive conclusion that although we can see it as true, we cannot mathematically prove the following proposition as true:

Proposition 1: Whether or not a prime $p$ divides an integer $n$ is independent of whether or not a prime $q \neq p$ divides the integer $n$.

We note that such an unprovable-but-intuitively-true conclusion makes a stronger assumption than that in Gödel’s similar claim for his arithmetical formula $[(\forall x)R(x)]$—whose Gödel-number is $17Gen\ r$—in [Go31], p.26(2). Stronger, since Gödel does not assume his proposition to be intuitively true, but shows that though the arithmetical formula with Gödel-number $17Gen\ r$ is not provable in his Peano Arithmetic $P$ yet, for any $P$-numeral $[n]$, the formula $[R(n)]$ whose Gödel-number is $Sb \left(r \begin{array}{c}17 \\ Z(n)\end{array}\right)$ is $P$-provable, and therefore meta-mathematically true under any well-defined Tarskian interpretation of $P$ (cf., [An16], Section 3.).

Expressed in computational terms (see [An16], Corollary 8.3), under any well-defined interpretation of $P$, Gödel’s formula $[R(x)]$ translates as an arithmetical relation, say $R'(x)$, such that $R'(n)$ is algorithmically verifiable, but not algorithmically computable, as always true over $N$, since $[\neg (\forall x)R(x)]$ is $P$-provable ([An16], Corollary 8.2).

We thus argue that a perspective which denies Proposition 1 is based on perceived barriers that reflect, and are peculiar to, only the argument that:

Theorem 2: There is no deterministic algorithm that, for any given $n$, and any given prime $p \geq 2$, will evidence that the probability $\mathbb{P}(p\ |\ n)$ that $p$ divides $n$ is $\frac{1}{p}$, and the probability $\mathbb{P}(p\not|\ n)$ that $p$ does not divide $n$ is $1 - \frac{1}{p}$.

Proof By a standard result in the Theory of Numbers ([Ste02], Chapter 2, p.9, Theorem 2.1, we cannot define a probability function for the probability that a random $n$ is prime over the probability space $(1, 2, 3, \ldots, )$.

(Compare with the informal argument in [HL23], pp.36-37.)

In other words, treating Theorem 2 as an absolute barrier does not admit the possibility—which has consequences for the resolution of outstanding problems in both the theory of numbers and computational complexity—that Proposition 1 is algorithmically verifiable, but not algorithmically computable, as true, since:

Theorem 3: For any given $n$, there is a deterministic algorithm that, given any prime $p \geq 2$, will evidence that the probability $\mathbb{P}(p\ |\ n)$ that $p$ divides $n$ is $\frac{1}{p}$, and the probability $\mathbb{P}(p\not|\ n)$ that $p$ does not divide $n$ is $1 - \frac{1}{p}$.

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