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

Abstract We define the residues for all and all such that if, and only if, is a divisor of . We then show that the joint probability of two unequal primes dividing any integer is the product . We conclude that the prime divisors of any integer are independent; and that the probability of being a prime is . The number of primes less than or equal to is thus given by . We further show that , and conclude that does not oscillate.

** The residues **

We begin by defining the residues for all and all as below:

**Definition 1:** where .

Since each residue cycles over the values , these values are all incongruent and form a complete system of residues ^{[1]} .

It immediately follows that:

**Lemma 1:** if, and only if, is a divisor of .

** The probability **

By the standard definition of the probability of an event , we conclude that:

**Lemma 2:** For any and any given integer , the probability that is , and the probability that is .

We note the standard definition:

**Definition 2:** Two events and are mutually independent for if, and only if, .

** The prime divisors of any integer are mutually independent**

We then have that:

**Lemma 3:** If and are two primes where then, for any , we have:

where and .

**Proof:** The numbers , where and , are all incongruent and form a complete system of residues ^{[2]} . Hence:

.

By Lemma 2:

.

The lemma follows.

If and in Lemma 3, so that both and are prime divisors of , we conclude by Definition 2 that:

**Corollary 1:** .

**Corollary 2:** .

**Theorem 1:** The prime divisors of any integer are mutually independent. ^{[3]}

** The probability that is a prime**

Since is a prime if, and only if, it is not divisible by any prime , it follows immediately from Lemma 2 and Lemma 3 that:

**Lemma 4:** For any , the probability of an integer being a prime is the probability that for any if .

**Lemma 5:** ^{[4]}.

** The Prime Number Theorem**

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

**Lemma 6:** .

This now yields the Prime Number Theorem:

**Theorem 2:** .

**Proof:** From Lemma 6 and Mertens’ Theorem that

^{[5]}:

it follows that:

The behaviour of as is then seen by differentiating the right hand side, where we note that :

Hence does not oscillate as .

**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 ; from which we conclude that integer factorising cannot be in the class 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.

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