Symmetric primes revisited

Carl Pomerance; Paul Pollack; William Banks

REVIEW 1 major objections 4 minor 14 references

Symmetric primes revisited

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves the count of symmetric primes up to x is at most π(x)/(log x)^η with η≈0.086, and that arbitrarily long consecutive-prime strings are pairwise symmetric.

desk verdict A short, elegant paper that settles the two main open problems on symmetric primes; the only serious soft spot is a load-bearing sieve estimate that is asserted but not proved. read the letter →

arxiv 1908.06161 v2 pith:7TBQLMGW submitted 2019-08-16 math.NT

William Banks , Paul Pollack , Carl Pomerance This is my paper · ORCID

classification math.NT MSC 11N0511N3611N3711A41

keywords symmetricprimesasymmetricgcdconditionprimetuplessievemethodsdivisorsumsconsecutivemultiplicationtableconstant

topics The Twin Prime Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Two old problems about symmetric primes — pairs of odd primes $\{p,q\}$ with $\gcd(p-1,q-1)=|p-q|$ — are settled here. First, the number $S(x)$ of symmetric primes up to $x$ is at most $\pi(x)/(\log x)^{\eta}\,(\log\log x)^{O(1)}$, where $\eta=1-(1+\log\log 2)/\log 2\approx 0.08607$; this proves the upper-bound conjecture from the 1996 paper that introduced the term. Second, for every $m\ge 2$ there is a block of $m$ consecutive primes in which every two primes form a symmetric pair, so in particular infinitely many symmetric primes exist. The interest is that symmetric primes are sparse but not too sparse, with the same exponent $\eta$ that governs the multiplication-table problem.

What carries the argument

The argument for Theorem 1.1 splits primes $p\le x$ according to the largest prime factor $r$ of $p-1$ and the number $\Omega(a)$ of prime factors of $a=(p-1)/r$. If $p$ is symmetric, some divisor $d$ of $a$ has one of $p\pm d$, $p\pm dr$ prime; this reduces the count to a uniform Brun-sieve estimate $R(x,d,m)$ for primes $r$ with $dmr+1$ prime and a shifted value also prime. Summing that estimate over pairs $d,m$ with few prime factors, using the divisor-sum bounds in [7], produces the extra $(\log x)^{-\eta}$ factor and the constant $\eta$. For Theorem 1.2, the machinery is a finite set $A$ with pairwise gcd equal to difference (an example for size four is $\{6,8,9,12\}$): if $an+1$ and $bn+1$ are prime for $a,b\in A$, then the two primes are symmetric. The bounded-gaps theorem guarantees infinitely many $n$ for which the $k$ forms $an+1$ contain $m$ primes, and a separate theorem on consecutive primes in admissible tuples converts those $m$ primes into a string of consecutive primes.

What would settle it

The most direct falsifier is a count: any sequence $x_j\to\infty$ with $S(x_j)/\pi(x_j)$ failing to decay at least as fast as $(\log x_j)^{-\eta}(\log\log x_j)^{O(1)}$ would disprove Theorem 1.1. Since finite data cannot settle this, the practical check is the proof's only unproved ingredient: verify the asserted uniform estimate $R(x,d,m)\le x/(dm(\log x)^3)(\log\log x)^{O(1)}$ for all $d,m$ with $dm<x^{1-1/\log\log x}$ and $\Omega(dm)<L$; an infinite family violating that estimate would remove the support of the upper bound, though not necessarily the theorem itself.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a two-part refinement of the 1996 results. Theorem 1.1 gives $S(x)\le \pi(x)/(\log x)^{\eta}(\log\log x)^{O(1)}$ for all large $x$, with $\eta=1-(1+\log\log 2)/\log 2$, improving the old upper bound whose exponent was only $0.027$ and matching the exponent conjectured earlier. Theorem 1.2 shows that for each $m\ge 2$ one can find $m$ consecutive primes $p_1<\dots<p_m$ with $\gcd(p_i-1,p_j-1)=|p_i-p_j|$ for all $i\ne j$, so every pair is symmetric. The proof of Theorem 1.2 constructs finite sets $A$ of integers with $\gcd(a,b)=|a-b|$ for distinct $a,b$ (for size four, $\{6,8,9,12\}$), turns the linear forms $an+1$ into primes via a bounded-gaps theorem for admissible tuples, and then applies a result on consecutive primes in admissible tuples to make the resulting primes consecutive. This answers the open infinitude problem in the strong form that symmetric primes occur in arbitrarily long consecutive clusters.

Load-bearing premise

The load-bearing premise is the uniform Brun-sieve estimate $R(x,d,m)\le x/(dm(\log x)^3)(\log\log x)^{O(1)}$, asserted without proof for all $d,m$ in the required range; if that estimate fails in the needed uniformity, the proof of Theorem 1.1 does not go through.

Editorial extensions

If this is right

If Theorem 1.1 is right, symmetric primes have natural density zero among primes: $S(x)/\pi(x)\to 0$ as $x\to\infty$, albeit slowly.
The upper-bound exponent $\eta$ matches the exponent conjectured in the 1996 paper, and the authors' heuristic predicts the matching lower bound $S(x)=\pi(x)/(\log x)^{\eta+o(1)}$.
Theorem 1.2 implies infinitely many symmetric primes, and in fact for every $m$ the graph on odd primes whose edges are symmetric pairs contains a $K_m$ with its vertices consecutive primes.
Because any symmetric pair with $p<q$ must satisfy $q<2p$, the graph contains no infinite complete subgraph even though it contains arbitrarily large finite ones.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the same $\eta$ appearing in the multiplication-table problem is not a coincidence: both estimates are controlled by divisor sums over integers with restricted numbers of prime factors, so a sharp form of one bound may transfer to the other.
A wide gap remains between the proved upper-bound exponent $0.086$ and the lower-bound exponent $49$ mentioned in the paper; improving the uniform sieve estimate or the divisor-sum step is the natural route to narrowing it.
The sets $A$ with $\gcd(a,b)=|a-b|$ are of independent combinatorial interest; studying the minimal size of such sets and the least $n$ for which the $an+1$ are prime would give computable lower bounds for the first $m$-consecutive symmetric cluster.
The graph questions in Section 5 suggest a testable extension: computing the connected component containing the prime 3 far beyond the paper's table could indicate whether symmetric primes form one giant component or many, and whether every connected component is finite.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Paul, this one is worth your time. Banks, Pollack, and Pomerance resolve both open problems from the 1996 Fletcher–Lindgren–Pomerance paper: they prove the conjectured power-of-log upper bound with the sharp exponent eta = 0.08607, and they show that symmetric primes occur in arbitrarily long runs of consecutive primes. That is a real advance, and the exposition is unusually clear for a paper this dense.

The proof of Theorem 1.1 is a clean three-case sieve argument. The decomposition into S1, S2, and the remaining middle case is standard, and the divisor-sum computation via Hall–Tenenbaum bounds is neat. Theorem 1.2 is the more surprising: combining Heath-Brown's gcd sets with Maynard–Tao and the Banks–Freiberg–Turnage-Butterbaugh theorem on consecutive primes is a genuinely new construction, and the admissibility check is clever.

The main soft spot is exactly what the stress-test flagged: the uniform Brun-type bound for R(x,d,m) in Section 2 is asserted as "Again by Brun's method" with no proof and no precise citation. This bound is load-bearing, because it supplies the extra (log x) factor that converts the divisor sum into the claimed upper bound. If it failed in the required uniformity, Theorem 1.1 would collapse. That said, the estimate is a standard three-linear-forms sieve problem, the range dm < x^{1-1/log log x} leaves log X at least (log x)/log log x, and the singular series is no larger than (log log x)^{O(1)}. I am fairly confident the bound is true and can be proved by adapting Halberstam–Richert, but the paper as written leaves it as an unstated lemma.

The reliance on Maynard–Tao and BFTB is appropriate; these are published external results, and the self-citation to [1] is legitimate since it is the exact tool needed. The computations in Section 4 are a nice sanity check, though they do not strongly discriminate between the old and new exponents.

Bottom line: the central results are very likely correct, and the proof is structurally sound modulo that one sieve estimate. This deserves a serious referee, who should ask the authors to supply a proof or reference for the Brun bound. I would accept it for peer review with that request.

Referee Report

1 major / 4 minor

Summary. The paper studies symmetric pairs of odd primes, defined by gcd(p−1, q−1) = |p−q|. It proves two main results. Theorem 1.1 gives the upper bound S(x) ≤ π(x)/(log x)^η (log log x)^{O(1)} for all large x, where η = 1 − (1 + log log 2)/log 2 = 0.08607..., improving the 1996 Fletcher–Lindgren–Pomerance upper bound and confirming the conjecture from that paper. Theorem 1.2 states that for every integer m ≥ 2 there is a string of m consecutive primes, any two of which form a symmetric pair, which implies the infinitude of symmetric primes. The proof of Theorem 1.1 splits primes according to P^+(p−1) and Ω(p−1), uses two Brun-type sieve estimates and Hall–Tenenbaum divisor sums, and reduces the main term to an exponential divisor-sum bound. The proof of Theorem 1.2 combines sets with the property gcd(a,b)=|a−b|, the Maynard–Tao theorem, and a theorem of Banks–Freiberg–Turnage-Butterbaugh on consecutive primes in admissible tuples.

Significance. If the proofs are correct, this is a substantial advance: it resolves the 1996 conjecture on the exponent in the upper bound for symmetric primes and answers the long-standing question of whether infinitely many symmetric primes exist, in the strong form of arbitrarily long consecutive-prime strings. The architecture of the proof is sensible: the S1/S2/remaining split is natural, the divisor-sum calculation is clean and yields exactly the exponent 2−η, and the use of Hall–Tenenbaum estimates is well matched to the problem. The paper also contains new computations of S(x) up to the 10^8-th prime. The main caveat is that the central upper bound relies on a uniform Brun-type estimate for R(x,d,m) that is asserted but not proved or precisely cited; this point is load-bearing and needs to be made explicit for the proof to be fully rigorous.

major comments (1)

[Section 2, estimate for R(x,d,m)] The estimate for R(x,d,m) is load-bearing for Theorem 1.1. Please add a proof or a precise citation with the required uniformity.

minor comments (4)

[Section 3, application of Theorem 3.4] Theorem 3.4 is stated for k = k_m, but it is subsequently applied with k ≥ k_m. The intended argument is clear, since a subset of size k_m of an admissible tuple remains admissible and preserves the gcd property, but the text should either state the theorem for all k ≥ k_m or explicitly pass to a subset of size k_m before applying it.
[Section 2, divisor-sum calculation] The quantity E is defined as the reciprocal sum of all primes and prime powers less than x, while the inner sums restrict d and m to be less than x. The inequality ∑_{ω(d)=i} 1/d ≤ E^i/i! is valid, but it would help the reader to note that using the full E is an upper bound and that the restriction to d < x is not used in the estimate.
[Section 3, Maynard–Tao attribution] The proof of Lemma 3.2 invokes a theorem of Maynard and 'Tao (unpublished)'; since part of this result is unpublished, the paper should provide a precise reference for the full statement used here, in addition to the citation to Maynard's paper.
[Section 4, Table 1] The sentence comparing the descent in S(p_n)/n with 'the main term in our upper bound' is informal; it would be clearer to state explicitly that the comparison is with (log p_n)^{−η}, as shown in the last column of Table 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the upper bound is a sieve calculation with fixed arithmetic constants, and the consecutive-primes theorem rests on external Maynard-Tao and Banks-Freiberg-Butterbaugh results that do not contain the symmetric-pair conclusion.

full rationale

Walking the derivation chain, Theorem 1.1 does not assume the 1996 bound it improves; the proof starts from Brun's method estimates and divisor-sum bounds from Halberstam-Richert and Hall-Tenenbaum. The constant eta is not fitted; it is the same arithmetic constant appearing in the multiplication table problem. The structural fact that a symmetric prime p=ar+1 has a divisor d|a with one of p+d, p-d, p+dr, p-dr prime is a geometric characterization from the earlier paper [4]; it is published prior work, it is not the S(x) bound, and it does not restate the theorem being proved. The uniform bound R(x,d,m) <= x/(dm(log x)^3)(log log x)^O(1) is asserted as an external sieve estimate, 'Again by Brun's method'; even if the paper leaves it unproved, an unverified technical input is a correctness risk, not circularity, because it does not reduce to the target conclusion. Theorem 1.2 invokes the Maynard-Tao theorem and the Banks-Freiberg-Butterbaugh theorem. The latter is a self-citation in the sense that one author overlaps, but it is an independent published theorem about consecutive primes in admissible tuples and never mentions symmetric pairs; it does not assume the present result. Lemma 3.2 gives an explicit reduction from symmetric primes to the sets A_k of Lemma 3.1, and the final step is a direct divisibility argument. No parameter is fitted and renamed as a prediction, no quantity is defined in terms of the target, and no uniqueness claim is imported from the authors' prior work to force a choice. Therefore the derivation is self-contained against external benchmarks and has no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's theorems are built entirely from published analytic number theory: de Bruijn's smooth-number estimate, Brun's sieve, Hall-Tenenbaum divisor sums, Heath-Brown's admissible sets, the Maynard-Tao theorem, and the BFTB consecutive-primes theorem. No numerical constant is fitted to data, and no new mathematical entity is introduced. The only nontrivial internal step is the d|a reduction in Section 2, which follows because r is prime; the sieve bound for R is asserted rather than proved.

assumptions (7)

standard math The number of n ≤ x with P^+(n) ≤ x^(1/log log x) is O(x/(log x)^2) (de Bruijn).
Used in Section 2 to dispose of S1(x).
domain assumption Uniform Brun-type sieve bound: for fixed a < x^(1-1/log log x), the count of primes r ≤ x/a with ar+1 prime is ≤ x/(a(log x)^2)(log log x)^O(1).
Invoked in Section 2; cited to Halberstam-Richert [6, Eq. (6.1)].
domain assumption Uniform Brun-type bound R(x,d,m) ≤ x/(dm(log x)^3)(log log x)^O(1).
Load-bearing sieve estimate in Section 2 for the final symmetric-prime count.
standard math Hall-Tenenbaum divisor sums: for L ≤ Ω(a) ≤ 1.9 log log x, sum 1/a ≤ (log x)^(1-η)(log log x)^O(1); for larger Ω(a), sum 1/a ≪ (log x)^0.69.
Used to sum the individual sieve bounds in Section 2; from [7, Theorem 08 and Exercise 05].
domain assumption For every k ≥ 2 there exists a k-element set A_k with gcd(a,b)=|a-b| for all distinct a,b (Heath-Brown).
Lemma 3.1, credited to [8,9,10], used to build the linear forms in Lemma 3.2.
domain assumption Maynard-Tao theorem (Theorem 3.3): admissible k-tuples of linear forms contain m primes for infinitely many n.
Deep external theorem used in proof of Lemma 3.2.
domain assumption Banks-Freiberg-Turnage-Butterbaugh theorem (Theorem 3.4): under admissibility and coprime g, some m shifted values are consecutive primes.
Deep external theorem used to pass from symmetric prime sets to consecutive primes; [1].

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Cite this review

Pith. "Pith review of Symmetric primes revisited." pith.science (2026). https://pith.science/paper/7TBQLMGW

@misc{pith2026190806161,
  author       = {Pith},
  title        = {Pith review of: Symmetric primes revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TBQLMGW}},
  note         = {Machine review of arXiv:1908.06161}
}

read the original abstract

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a symmetric pair. In the present paper, that theorem is improved to what is likely to be the best possible result. We also establish that there exist infinitely many symmetric pairs of primes. In fact, we show that for every integer m at least 2 there is a string of m consecutive primes, any two of which form a symmetric pair.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

W. D. Banks, T. Freiberg, and C. Turnage-Butterbaugh, Con secutive primes in tuples. Acta Arith. 167 (2015), 261–266

work page 2015
[2]

N. G. de Bruijn, On the number of positive integers ⩽ x and free of prime factors > y . Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50–60

work page 1951
[3]

Chow and C

S. Chow and C. Pomerance, Triangles with prime hypotenuse . Research in Number Theory 3 (2017), Art. 21, 10 pp., http://rdcu.be/wApe

work page 2017
[4]

Fletcher, W

P. Fletcher, W. Lindgren, and C. Pomerance, Symmetric and asymmetric primes. J. Number Theory 58 (1996), 89–99

work page 1996
[5]

Ford, The distribution of integers with a divisor in a gi ven interval

K. Ford, The distribution of integers with a divisor in a gi ven interval. Ann. of Math. (2) 168 (2008), 367–433. 6 WILLIAM BANKS, PAUL POLLACK, AND CARL POMERANCE

work page 2008
[6]

Halberstam and H.-E

H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4. Academic Press [A subsidiary of Harcourt Brace Jovano vich, Publishers], London–New York, 1974

work page 1974
[7]

R. R. Hall and G. Tenenbaum, Divisors, Cambridge University Press, Cambridge, 1988

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[8]

D. R. Heath-Brown, The divisor function at consecutive in tegers. Mathematika 31 (1984), 141–149

work page 1984

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[9]

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C. A. Spiro, The frequency with which an integral-valued, prime-indepe ndent, multiplicative or additive function of n divides a polynomial function of n, Ph.D. Thesis, Univ. Illinois, Urbana, IL 1981. Department of Mathematics, University of Missouri, Columbi a, MO 65211, USA...

1981

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

W. D. Banks, T. Freiberg, and C. Turnage-Butterbaugh, Con secutive primes in tuples. Acta Arith. 167 (2015), 261–266

work page 2015

[2] [2]

N. G. de Bruijn, On the number of positive integers ⩽ x and free of prime factors > y . Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50–60

work page 1951

[3] [3]

Chow and C

S. Chow and C. Pomerance, Triangles with prime hypotenuse . Research in Number Theory 3 (2017), Art. 21, 10 pp., http://rdcu.be/wApe

work page 2017

[4] [4]

Fletcher, W

P. Fletcher, W. Lindgren, and C. Pomerance, Symmetric and asymmetric primes. J. Number Theory 58 (1996), 89–99

work page 1996

[5] [5]

Ford, The distribution of integers with a divisor in a gi ven interval

K. Ford, The distribution of integers with a divisor in a gi ven interval. Ann. of Math. (2) 168 (2008), 367–433. 6 WILLIAM BANKS, PAUL POLLACK, AND CARL POMERANCE

work page 2008

[6] [6]

Halberstam and H.-E

H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4. Academic Press [A subsidiary of Harcourt Brace Jovano vich, Publishers], London–New York, 1974

work page 1974

[7] [7]

R. R. Hall and G. Tenenbaum, Divisors, Cambridge University Press, Cambridge, 1988

work page 1988

[8] [8]

D. R. Heath-Brown, The divisor function at consecutive in tegers. Mathematika 31 (1984), 141–149

work page 1984