REVIEW
Sieve methods and the twin prime conjecture
General Mathematics · Submitted 5 Sep 2019 · 15 references
Pith's verdict: Attempted proof of twin primes that fails on an unproved periodicity step; the specific inequality is original but the main induction does not go through.
T0 · AI review
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Referee report
The paper
The paper claims
The paper claims to prove the twin prime conjecture by proving a floor-function inequality for the twin-prime counting function.
The reading
What carries the argument
The central object is the inclusion-exclusion sieve S(x,n) of equation (2.5), which counts integers up to x that are coprime to the first n primes. For twin primes the paper works with the paired residue set R(x,n) = {t<k : neither 6t-1 nor 6t+1 is divisible by any prime p_s, 3<=s<=n}, whose size is compared with half of S(x,n). The argument's engine is the assertion that S(x,n) is cyclic over every interval of length equal to the product of the first n primes, so that checking x only up to that product in the n=70 base case is claimed to determine the behavior for all larger x; comparing x/(2(n+1)) with |R(x,n)| gives the induction.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is Theorem 1.3: there exists an integer N2(1), identified as 20, such that for every n>=20, floor(p_{n+1}^2/(2(n+1))) <= pi_2(p_{n+1}^2). Since the floor sequence grows without bound, the inequality implies that the twin-prime count at p_{n+1}^2 must also grow without bound, so there are infinitely many twin primes. The paper also proves an analogous prime-counting bound for pi(p_{n+1}^2) and shows that a standard heuristic on twin-prime density would imply the stronger Conjecture 1.2 for every positive integer a.
Load-bearing premise
The proof's load-bearing premise is that the sieve S(x,n) in equation (2.5) repeats periodically with period equal to the product of the first n primes; the paper asserts this cyclicity without proof or citation, and the entire induction depends on it.
Editorial extensions
If this is right
- If Theorem 1.3 is correct, the twin prime conjecture follows immediately because the sequence floor(p_{n+1}^2/(2(n+1))) is unbounded.
- The claimed threshold N2(1)=20 means the inequality is asserted to hold for every n>=20, with n=20 through 69 checked independently and n=70 onward covered by the induction.
- The stronger Conjecture 1.2, if proved, would give eventual lower bounds a*p_{n+1}^2/(2(n+1)) on pi_2(p_{n+1}^2) for every positive integer a; the paper only proves a=1.
- The paper says a standard twin-prime density conjecture would imply Conjecture 1.2 in full, so the a=1 result is a weaker consequence of the same comparison.
Reading between the lines
- The proof's cyclicity assumption is doing all the heavy lifting; until it is proved, the finite tables in the paper are evidence only for the checked values, not for all larger x.
- If the cyclicity assumption could be proved, the same finite-check-plus-induction strategy would likely extend to other fixed patterns of primes, not just pairs separated by 2.
- The quantitative content of the claimed bound is extremely weak: p_{n+1}^2/(2(n+1)) grows like n log^2 n, whereas the standard twin-prime density heuristic predicts pi_2(p_{n+1}^2) grows like n^2, so even a correct proof would not approach the conjectured counts.
Original abstract
For $n \geq 3,$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $[ \; ]$ denote the floor or greatest integer function. For a positive integer $m,$ let $\pi_2(m)$ denote the number of twin primes not exceeding $m.$ The twin prime conjecture states that there are infinitely many prime numbers $p$ such that $p+2$ is also prime. In this paper we state a conjecture to the effect that given any integer $a>0$ there exists an integer $N_2(a)$ such that $$ \left[\frac{ap^2_{n+1}}{2(n+1)} \right] \leq \pi_2\left(p^2_{n+1} \right) $$ for all $n \geq N_2(a)$ and prove the conjecture in the case $a=1.$ This, in turn, establishes the twin prime conjecture.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
For $n \geq 3,$ let $ p_n $ denote the $n^{\rm th}$ prime number. Let $[ \; ]$ denote the floor or greatest integer function. For a positive integer $m,$ let $\pi_2(m)$ denote the number of twin primes not exceeding $m.$ The twin prime conjecture states that there are infinitely many prime numbers $p$ such that $p+2$ is also prime. In this paper we state a conjecture to the effect that given any integer $a>0$ there exists an integer $N_2(a)$ such that $$ \left[\frac{ap^2_{n+1}}{2(n+1)} \right] \leq \pi_2\left(p^2_{n+1} \right) $$ for all $n \geq N_2(a)$ and prove the conjecture in the case $a=1.$ This, in turn, establishes the twin prime conjecture.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For $n \geq 3,$ let $ p_n $ denote the $n^{\rm th}$ prime number. | background 1.00 | 0.12 | 0.01 | asserted 0.75 |
| s1 | Let $[ \; ]$ denote the floor or greatest integer function. | background 1.00 | 0.06 | 0.01 | asserted 0.81 |
| s2 | For a positive integer $m,$ let $\pi_2(m)$ denote the number of twin primes not exceeding $m.$ The twin prime conjecture states that there are infinitely many prime numbers $p$ such that $p+2$ is also prime. | background 1.00 | 0.06 | 0.04 | asserted 0.73 |
| s3 | In this paper we state a conjecture to the effect that given any integer $a>0$ there exists an integer $N_2(a)$ such that $$ \left[\frac{ap^2_{n+1}}{2(n+1)} \right] \leq \pi_2\left(p^2_{n+1} \right) $$ for all $n \geq N_2(a)$ and prove the conjecture in the case $a=1.$ This, in turn, establishes the twin prime conjecture. | contribution 0.90 | 0.93 | 0.94 | asserted 0.64 |
What was examined, and what was not
- arXiv listing abstract v13
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- P abstract
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References
How the literature uses it
1 arXiv paper cite this in 2 sentences
Papers citing it
Builds on it 1 paper
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Our arguments are an extension or generalization of the arguments developed in [ref] .
A proof of cases of de Polignac's conjecture · Introduction and main result · 0.99
Cites it as background 1 paper
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In $1849,$ Alphonse de Polignac [ref] conjectured that for every even number $k,$ there are infinitely many pairs of prime numbers $p$ and $p'$ such that $p'-p = k.$ The case $k=2$ is the well known twin prime conjecture, which is considered in [ref] .
A proof of cases of de Polignac's conjecture · Introduction and main result · 0.98
Discussion
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