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Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

1. Since, by the Prime Number Theorem, the number of primes \leq \sqrt n is O(\frac{\sqrt n}{log_{_{e}}\sqrt n}), it would follow that determining a factor of n requires at least one logical operation for each prime \leq \sqrt n, and therefore cannot be done in polynomial time—whence P \neq NPIF whether or not a prime p divides an integer n were independent of whether or not a prime q \neq p divides the integer n.

2. Currently, conventional approaches to determining the computational complexity of Integer Factorising apparently appeal critically to the belief that:

(i) either—explicitly (see here)—that whether or not a prime p divides an integer n is not independent of whether or not a prime q \neq p divides the integer n;

(ii) or—implicitly (since the problem is yet open)—that a proof to the contrary must imply that if P(n\ is\ a\ prime) is the probability that n is a prime, then \sum_{_{i = 1}}^{^{\infty}} P(i\ is\ a\ prime) = 1.

3. If so, then conventional approaches seem to conflate the two probabilities:

(i) The probability P(a) of selecting a number that has the property of being prime from a given set S of numbers;

Example 1: I have a bag containing 100 numbers in which there are twice as many composites as primes. What is the probability that the first number you blindly pick from it is a prime. This is the basis for setting odds in games such as roulette.

(ii) The probability P(b) of determining that a given integer n is prime.

Example 2: I give you a 5-digit combination lock along with a 10-digit number n. The lock only opens if you set the combination to a proper factor of n which is greater than 1. What is the probability that the first combination you try will open the lock. This is the basis for RSA encryption, which provides the cryptosystem used by many banks for securing their communications.

4. In case 3(i), if the precise proportion of primes to non-primes in S is definable, then clearly P(a) too is definable.

However if S is the set N of all integers, and we cannot define a precise ratio of primes to composites in N, but only an order of magnitude such as O(\frac{1}{log_{_{e}}n}), then equally obviously P(a) cannot be defined in N (see Chapter 2, p.9, Theorem 2.1, here).

5. In case 3(ii) the following paper proves P(b) = \frac{1}{\pi(\sqrt{n})}, since it shows that whether or not a prime p divides a given integer n is independent of whether or not a prime q \neq p divides n:

Why Integer Factorising cannot be polynomial time

Not only does it immediately follow that P \neq NP (see here), but we further have that \pi(n) \approx n.\prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}}), with a binomial standard deviation. Hence, even though we cannot define the probability P(n\ is\ a\ prime) of selecting a number from the set N of all natural numbers that has the property of being prime, \prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}}) can be treated as the de facto probability that a given n is prime, with all its attended consequences for various prime-counting functions and the Riemann Zeta function (see here).

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

This argument laid the foundation for this investigation. See also this arXiv preprint, and this broader update.

Abstract We define the residues r_{i}(n) for all n \geq 2 and all i \geq 2 such that r_{i}(n) = 0 if, and only if, i is a divisor of n. We then show that the joint probability \mathbb{P}(p_{i} | n\ \cap\ p_{j} | n) of two unequal primes p_{i},\ p_{j} dividing any integer n is the product \mathbb{P}(p_{i} | n).\mathbb{P}(p_{j} | n). We conclude that the prime divisors of any integer n are independent; and that the probability \mathbb{P}(n \in \{p\}) of n being a prime p is \prod_{i = 1}^{\sqrt{n}}(1 - 1/p_{i}) \sim 2e^{-\gamma}/log_{e}\ n. The number of primes less than or equal to n is thus given by \pi(n) \approx \sum_{j = 1}^{n}\prod_{i = 1}^{\pi(\sqrt{j})}(1 - 1/p_{i}). We further show that ((\pi(x).log_{e}\ x)/x)' = o(1/x), and conclude that (\pi(x).log_{e}\ x)/x does not oscillate.

\S 1 The residues r_{i}(n)

We begin by defining the residues r_{i}(n) for all n \geq 2 and all i \geq 2 as below:

Definition 1: n + r_{i}(n) \equiv 0\ (mod\ i) where i > r_{i}(n) \geq 0.

Since each residue r_{i}(n) cycles over the i values (i-1, i-2, \ldots, 0), these values are all incongruent and form a complete system of residues [1] mod\ i.

It immediately follows that:

Lemma 1: r_{i}(n) = 0 if, and only if, i is a divisor of n.

\S 2 The probability \mathbb{P}(e)

By the standard definition of the probability \mathbb{P}(e) of an event e, we conclude that:

Lemma 2: For any n \geq 2,\ i \geq 2 and any given integer i > u \geq 0, the probability \mathbb{P}(r_{i}(n) = u) that r_{i}(n) = u is 1/i, and the probability \mathbb{P}(r_{i}(n) \neq u) that r_{i}(n) \neq u is 1 - 1/i.

We note the standard definition:

Definition 2: Two events e_{i} and e_{j} are mutually independent for i \neq j if, and only if, \mathbb{P}(e_{i}\ \cap\ e_{j}) = \mathbb{P}(e_{i}).\mathbb{P}(e_{j}).

\S 3 The prime divisors of any integer n are mutually independent

We then have that:

Lemma 3: If p_{i} and p_{j} are two primes where i \neq j then, for any n \geq 2, we have:

\mathbb{P}((r_{p_{_{i}}}(n) = u) \cap (r_{p_{_{j}}}(n) = v)) = \mathbb{P}(r_{p_{_{i}}}(n) = u).\mathbb{P}(r_{p_{_{j}}}(n) = v)

where p_{i} > u \geq 0 and p_{j} > v \geq 0.

Proof: The p_{i}.p_{j} numbers v.p_{i} + u.p_{j}, where p_{i} > u \geq 0 and p_{j} > v \geq 0, are all incongruent and form a complete system of residues [2] mod\ (p_{i}.p_{j}). Hence:

\mathbb{P}((r_{p_{_{i}}}(n) = u) \cap (r_{p_{_{j}}}(n) = v)) = 1/p_{i}.p_{j}.

By Lemma 2:

\mathbb{P}(r_{p_{_{i}}}(n) = u).\mathbb{P}(r_{p_{_{j}}}(n) = v) = (1/p_{i})(1/p_{j}).

The lemma follows. \Box

If u = 0 and v = 0 in Lemma 3, so that both p_{i} and p_{j} are prime divisors of n, we conclude by Definition 2 that:

Corollary 1: \mathbb{P}((r_{p_{_{i}}}(n) = 0) \cap (r_{p_{_{j}}}(n) = 0)) = \mathbb{P}(r_{p_{_{i}}}(n) = 0).\mathbb{P}(r_{p_{_{j}}}(n) = 0).

Corollary 2: \mathbb{P}(p_{i} | n\ \cap\ p_{j} | n) = \mathbb{P}(p_{i} | n).\mathbb{P}(p_{j} | n).

Theorem 1: The prime divisors of any integer n are mutually independent. [3]

\S 4 The probability that n is a prime

Since n is a prime if, and only if, it is not divisible by any prime p \leq \sqrt{n}, it follows immediately from Lemma 2 and Lemma 3 that:

Lemma 4: For any n \geq 2, the probability \mathbb{P}(n \in \{p\}) of an integer n being a prime p is the probability that r_{p_{_{i}}}(n) \neq 0 for any 1 \leq i \leq k if p_{k}^{2} \leq n < p_{k+1}^{2}. \Box

Lemma 5: \mathbb{P}(n \in \{p\}) = \prod_{i = 1}^{\pi(\sqrt{n})}(1 - 1/p_{i}) \sim 2e^{-\gamma}/log_{e}\ n [4]. \Box

\S 5 The Prime Number Theorem

The number of primes less than or equal to n is thus given by:

Lemma 6: \pi(n) \approx \sum_{j = 1}^{n}\prod_{i = 1}^{\pi(\sqrt{j})}(1 - 1/p_{i}). \Box

This now yields the Prime Number Theorem:

Theorem 2: \pi(x) \sim x/log_{e}\ x.

Proof: From Lemma 6 and Mertens’ Theorem that
[5]:

\prod_{p \leq x}(1 - 1/p_{i}) = e^{o(1)}.e^{-\gamma}/log_{e}\ x

it follows that:

(\pi(n).log_{e}\ n)/n \approx (log_{e}\ n/n)\sum_{j = 1}^{n}\prod_{i = 1}^{\pi(\sqrt{j})}(1 - 1/p_{i})

(\pi(n).log_{e}\ n)/n \approx (log_{e}\ n/n)\sum_{j = 1}^{n}(2.e^{o(1)}.e^{-\gamma})/log_{e}\ n

(\pi(x).log_{e}\ x)/x \sim c.(log_{e}\ x/x)\int_{2}^{x}(1/log_{e}\ t).dt

(\pi(x).log_{e}\ x)/x \sim c.(log_{e}\ x/x)[log_{e}.log_{e}\ t + log_{e}\ t + \sum_{2}^{\infty}(log_{e}\ t)^{k}/(k. k!)]_{2}^{^{x}}

(\pi(x).log_{e}\ x)/x \sim c.(log_{e}\ x/x)[log_{e}.log_{e}\ x + log_{e}\ x + \sum_{2}^{\infty}(log_{e}\ x)^{k}/(k. k!)]

(\pi(x).log_{e}\ x)/x \sim c.(log_{e}\ x/x).f(log_{e}\ x)

The behaviour of (\pi(x).log_{e}\ x)/x as x \rightarrow \infty is then seen by differentiating the right hand side, where we note that f(log_{e}\ x)' = 1/log_{e}\ x:

(c.(log_{e}\ x/x).f(log_{e}\ x))' = c/x + c.f(log_{e}\ x).(1/x^{2} - log_{e}\ x/x^{2})

(c.(log_{e}\ x/x).f(log_{e}\ x))' = c/x + (c.f(log_{e}\ x).(1 - log_{e}\ x))/x^{2}

(c.(log_{e}\ x/x).f(log_{e}\ x))' = o(1/x)

Hence (\pi(x).log_{e}\ x)/x does not oscillate as x \rightarrow \infty. \Box

Acknowledgements

I am indebted to my erstwhile classmate, Professor Chetan Mehta, for his unqualified encouragement and support for my scholarly pursuits over the past fifty years; most pertinently for his patiently critical insight into the required rigour without which the argument of this 1964 investigation would have remained in the informal universe of seemingly self-evident truths.

References

HW60 G. H. Hardy and E. M. Wright. 1960. An Introduction to the Theory of Numbers 4th edition. Clarendon Press, Oxford.

Ti51 E. C. Titchmarsh. 1951. The Theory of the Riemann Zeta-Function. Clarendon Press, Oxford.

Notes

Return to 1: HW60, p.49.

Return to 2: HW60, p.52, Theorem 59.

Return to 3: In the previous post we have shown how it immediately follows from Theorem 1 that integer factorising is necessarily of order O(n/log_{e}\ n); from which we conclude that integer factorising cannot be in the class P of polynomial-time algorithms.

Return to 4: By HW60, p.351, Theorem 429, Mertens’ Theorem.

Return to 5: By the argument in Ti51, p.59, eqn.(3.15.2).

Return to 6: HW60, p.9, Theorem 7.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

Follow me on Academia.edu

This argument laid the foundation for this later post and this investigation.

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

Abstract: We show the joint probability \mathbb{P}(p_{i} | n\ \cap\ p_{j} | n) that two unequal primes p_{i},\ p_{j} divide any integer n is the product \mathbb{P}(p_{i} | n).\mathbb{P}(p_{j} | n). We conclude that the prime divisors of any integer n are independent; and that Integer Factorising is necessarily of the order O(n/log_{e}\ n).

\S 1 The residues r_{i}(n)

We define the residues r_{i}(n) for all n \geq 2 and all i \geq 2 as below:

Definition 1: n + r_{i}(n) \equiv 0\ (mod\ i) where i > r_{i}(n) \geq 0.

Since each residue r_{i}(n) cycles over the i values (i-1, i-2, \ldots, 0), these values are all incongruent and form a complete system of residues [1] mod\ i.

We note that:

Lemma 1: r_{i}(n) = 0 if, and only if, i is a divisor of n.

\S 2 The probability \mathbb{P}(e)

By the standard definition of the probability [2] \mathbb{P}(e) of an event e, we then have that:

Lemma 2: For any n \geq 2,\ i \geq 2 and any given integer i > u \geq 0, the probability \mathbb{P}(r_{i}(n) = u) that r_{i}(n) = u is 1/i, and the probability \mathbb{P}(r_{i}(n) \neq u) that r_{i}(n) \neq u is 1 - 1/i.

We note the standard definition [3]:

Definition 2: Two events e_{i} and e_{j} are mutually independent for i \neq j if, and only if, \mathbb{P}(e_{i}\ \cap\ e_{j}) = \mathbb{P}(e_{i}).\mathbb{P}(e_{j}).

\S 3 The prime divisors of any integer n are mutually independent

We then have that:

Lemma 3: If p_{i} and p_{j} are two primes where i \neq j then, for any n \geq 2, we have:

\mathbb{P}((r_{p_{_{i}}}(n) = u) \cap (r_{p_{_{j}}}(n) = v)) = \mathbb{P}(r_{p_{_{i}}}(n) = u).\mathbb{P}(r_{p_{_{j}}}(n) = v)

where p_{i} > u \geq 0 and p_{j} > v \geq 0.

Proof: The p_{i}.p_{j} numbers v.p_{i} + u.p_{j}, where p_{i} > u \geq 0 and p_{j} > v \geq 0, are all incongruent and form a complete system of residues [4] mod\ (p_{i}.p_{j}). Hence:

\mathbb{P}((r_{p_{_{i}}}(n) = u) \cap (r_{p_{_{j}}}(n) = v)) = 1/p_{i}.p_{j}.

By Lemma 2:

\mathbb{P}(r_{p_{_{i}}}(n) = u).\mathbb{P}(r_{p_{_{j}}}(n) = v) = (1/p_{i})(1/p_{j}).

The lemma follows. \Box

If u = 0 and v = 0 in Lemma 3, so that both p_{i} and p_{j} are prime divisors of n, we conclude by Definition 2 that:

Corollary 1: \mathbb{P}((r_{p_{_{i}}}(n) = 0) \cap (r_{p_{_{j}}}(n) = 0)) = \mathbb{P}(r_{p_{_{i}}}(n) = 0).\mathbb{P}(r_{p_{_{j}}}(n) = 0).

Corollary 2: \mathbb{P}(p_{i} | n\ \cap\ p_{j} | n) = \mathbb{P}(p_{i} | n).\mathbb{P}(p_{j} | n).

Theorem 1: The prime divisors of any integer n are mutually independent.

Since n is a prime if, and only if, it is not divisible by any prime p \leq \sqrt{n} we may, without any loss of generality, take integer factorising to mean determining at least one prime factor p \leq \sqrt{n} of any given n \geq 2.

\S 4 Integer Factorising is not in P

It then immediately follows from Theorem 1 that:

Corollary 3: Integer Factorising is not in P.

Proof: We note that any computational process to identify a prime divisor of n \geq 2 must necessarily appeal to a logical operation for identifying such a factor.

Since n may be the square of a prime, it follows from Theorem 1 that we necessarily require at least one logical operation for each prime p \leq \sqrt{n} in order to logically identify a prime divisor of n.

Moreover, since the number of such primes is of the order O(n/log_{e}\ n), any deterministic algorithm that always computes a prime factor of n cannot be polynomial-time—i.e. of order O((log_{e}\ n)^{c}) for any c—in the length of the input n.

The corollary follows if P is the set of such polynomial-time algorithms. \Box

Acknowledgements

I am indebted to my erstwhile classmate, Professor Chetan Mehta, for his unqualified encouragement and support for my scholarly pursuits over the past fifty years; most pertinently for his patiently critical insight into the required rigour without which the argument of this 1964 investigation would have remained in the informal universe of seemingly self-evident truths.

References

GS97 Charles M. Grinstead and J. Laurie Snell. 1997. Introduction to Probability. Second Revised Edition, 1997, American Mathematical Society, Rhode Island, USA.

HW60 G. H. Hardy and E. M. Wright. 1960. An Introduction to the Theory of Numbers 4th edition. Clarendon Press, Oxford.

Ko56 A. N. Kolmogorov. 1933. Foundations of the Theory of Probability. Second English Edition. Translation edited by Nathan Morrison. 1956. Chelsea Publishing Company, New Yourk.

An05 Bhupinder Singh Anand. 2005. Three Theorems on Modular Sieves that suggest the Prime Difference is O(\pi(p(n)^{1/2})). Private investigation.

Notes

Return to 1: HW60, p.49.

Return to 2: Ko56, Chapter I, Section 1, Axiom III, p.2; see also GS97, Chapter 1, Section 1.2, Definition 1.2, p.19.

Return to 3: Ko56, Chapter VI, Section 1, Definition 1, pg.57 and Section 2, pg.58; see also GS97, Chapter 4, Section 4.1, Theorem 4.1, pg.140.

Return to 4: HW60, p.52, Theorem 59.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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