Finding narrow admissible tuples

For any natural number [math]\displaystyle{ k_0 }[/math], an admissible [math]\displaystyle{ k_0 }[/math]-tuple is a finite set [math]\displaystyle{ {\mathcal H} }[/math] of integers of cardinality [math]\displaystyle{ k_0 }[/math] which avoids at least one residue class modulo [math]\displaystyle{ p }[/math] for each prime [math]\displaystyle{ p }[/math]. (Note that one only needs to check those primes [math]\displaystyle{ p }[/math] of size at most [math]\displaystyle{ k_0 }[/math], so this is a finitely checkable condition.) Let [math]\displaystyle{ H(k_0) }[/math] denote the minimal diameter [math]\displaystyle{ \max {\mathcal H} - \min {\mathcal H} }[/math] of an admissible [math]\displaystyle{ k_0 }[/math]-tuple. As part of the Polymath8 project, we would like to find as good an upper bound on [math]\displaystyle{ H(k_0) }[/math] as possible for given values of [math]\displaystyle{ k_0 }[/math]. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size [math]\displaystyle{ k_0 \log k_0 + k_0 }[/math] for [math]\displaystyle{ k_0 }[/math] in the range of interest.

Upper bounds

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo [math]\displaystyle{ p }[/math] from an interval for a lot of primes [math]\displaystyle{ p }[/math]. Examples of sieves, in roughly increasing order of efficiency, are listed below.

Zhang sieve

The Zhang sieve uses the tuple

[math]\displaystyle{ {\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\} }[/math]

where [math]\displaystyle{ m }[/math] is taken to optimize the diameter [math]\displaystyle{ p_{m+k_0}-p_{m+1} }[/math] while staying admissible (in practice, this basically means making [math]\displaystyle{ m }[/math] as small as possible). Certainly any [math]\displaystyle{ m }[/math] with [math]\displaystyle{ p_{m+1} \gt k_0 }[/math] works; in particular, one can just take [math]\displaystyle{ {\mathcal H} }[/math] to be the first [math]\displaystyle{ k_0 }[/math] primes past [math]\displaystyle{ k_0 }[/math], but this is not optimal. Applying the prime number theorem then gives the upper bound [math]\displaystyle{ H \leq (1+o(1)) k_0\log k_0 }[/math].

Hensley-Richards sieve

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

[math]\displaystyle{ {\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\} }[/math]

where m is again optimised to minimize the diameter while staying admissible.

Asymmetric Hensley-Richards sieve

The asymmetric Hensley-Richard sieve uses the tuple

[math]\displaystyle{ {\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\} }[/math]

where [math]\displaystyle{ i }[/math] is an integer and [math]\displaystyle{ i,m }[/math] are optimised to minimize the diameter while staying admissible.

Schinzel sieve

Given [math]\displaystyle{ 0\lt y\lt z\lt x }[/math], the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval [math]\displaystyle{ [1,x] }[/math] by [math]\displaystyle{ 1 \bmod p }[/math] for primes [math]\displaystyle{ p \le y }[/math] and by [math]\displaystyle{ 0\bmod p }[/math] for primes [math]\displaystyle{ y \lt p \le z }[/math]. Provided that [math]\displaystyle{ z }[/math] is large enough ([math]\displaystyle{ z=k_0 }[/math] clearly suffices), the first [math]\displaystyle{ k_0 }[/math] survivors form an admissible [math]\displaystyle{ k_0 }[/math]-tuple (but not necessarily the narrowest one in the interval). The case [math]\displaystyle{ y=1 }[/math] corresponds to a sieve of Eratosthenes; if one minimizes [math]\displaystyle{ z }[/math] and takes the first [math]\displaystyle{ k_0 }[/math] survivors greater than 1, this yields the same admissible [math]\displaystyle{ k_0 }[/math] tuple as Zhang, with the minimal possible value of [math]\displaystyle{ m }[/math].

Shifted Schinzel sieve

As a generalization of the Schinzel sieve, one may instead sieve shifted intervals [math]\displaystyle{ [s,s+x] }[/math]. This is effectively equivalent to sieving the interval [math]\displaystyle{ [0,x] }[/math] of the residue classes [math]\displaystyle{ -s\ \bmod\ p }[/math] for primes [math]\displaystyle{ p\le y }[/math] and [math]\displaystyle{ 1-s\ \bmod\ p }[/math] for primes [math]\displaystyle{ y\lt p\le z }[/math].

Greedy sieve

Within a given interval, one sieves a single residue class [math]\displaystyle{ a \bmod p }[/math] for increasing primes [math]\displaystyle{ p=2,3,5,\ldots }[/math], with [math]\displaystyle{ a }[/math] chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize [math]\displaystyle{ a\in[0,p-1] }[/math], maximize [math]\displaystyle{ a\in [0,p-1] }[/math], minimize [math]\displaystyle{ |a-\lfloor p/2\rfloor| }[/math], or randomly. If not all residue classes modulo [math]\displaystyle{ p }[/math] are occupied by survivors, then [math]\displaystyle{ a }[/math] will be chosen so that no survivors are sieved. This necessarily occurs once [math]\displaystyle{ p }[/math] exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ {\mathcal H} }[/math] among the survivors (if there are fewer than [math]\displaystyle{ k_0 }[/math] survivors, retry with a wider interval).

Greedy-greedy sieve

Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on [math]\displaystyle{ [s,\ s+x] }[/math] using [math]\displaystyle{ y = 2 }[/math] and [math]\displaystyle{ z = \sqrt{x} }[/math] and then continuing in a greedy fashion, as proposed by Sutherland. One first optimizes the shift value [math]\displaystyle{ s }[/math] over some larger interval (e.g. [math]\displaystyle{ [-k_0\log\ k_0,\ k_0\log\ k_0] }[/math]) and then continues the sieving over primes [math]\displaystyle{ p \gt z }[/math] greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in [math]\displaystyle{ [0,\ p-1] }[/math]).

Seeded greedy sieve

Given an initial sequence [math]\displaystyle{ {\mathcal S} }[/math] that is known to contain an admissible [math]\displaystyle{ k_0 }[/math]-tuple, one can apply greedy sieving to the minimal interval containing [math]\displaystyle{ {\mathcal S} }[/math] until an admissible sequence of survivors remains, and then choose the narrowest [math]\displaystyle{ k_0 }[/math]=tuple it contains. The sieving methods above can be viewed as the special case where [math]\displaystyle{ {\mathcal S} }[/math] is the set of integers in some interval. The main difference is that the choice of [math]\displaystyle{ {\mathcal S} }[/math] affects when ties occur and how they are broken with greedy sieving. One approach is to take [math]\displaystyle{ {\mathcal S} }[/math] to be the union of two [math]\displaystyle{ k_0 }[/math]-tuples that lie in roughly the same interval (see Iterated merging) below.

Iterated merging

Given an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal{H}_1 }[/math], one can attempt to improve it using an iterated merging approach suggested by Castryck. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal{H}_2 }[/math] in roughly the same interval as [math]\displaystyle{ \mathcal{H}_1 }[/math], then performs a randomized greedy sieve using the seed set [math]\displaystyle{ \mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2 }[/math] to obtain an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal{H}_3 }[/math]. If [math]\displaystyle{ \mathcal{H}_3 }[/math] is narrower than [math]\displaystyle{ \mathcal{H}_2 }[/math], replace [math]\displaystyle{ \mathcal{H}_2 }[/math] with [math]\displaystyle{ \mathcal{H}_3 }[/math], otherwise try again with a new [math]\displaystyle{ \mathcal{H}_3 }[/math]. Eventually the diameter of [math]\displaystyle{ \mathcal{H}_2 }[/math] will become less than or equal to that of [math]\displaystyle{ \mathcal{H}_1 }[/math]. As long as [math]\displaystyle{ \mathcal{H}_1\ne \mathcal{H}_2 }[/math], one can continue to attempt to improve [math]\displaystyle{ \mathcal{H}_2 }[/math], but in practice one stops after some number of retries.

As described by Sutherland, one can then replace [math]\displaystyle{ \mathcal{H}_1 }[/math] with [math]\displaystyle{ \mathcal{H}_2 }[/math] and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing [math]\displaystyle{ \mathcal{H}_2 }[/math], which is typically made wider than the minimal interval containing [math]\displaystyle{ \mathcal{H}_1 }[/math] by a small factor [math]\displaystyle{ \delta }[/math] on each side (Sutherland suggests [math]\displaystyle{ \delta = 0.0025 }[/math]), and the number of failed attempts allowed while attempting to impove [math]\displaystyle{ \mathcal{H}_2 }[/math].

Eventually this process will tend to converge to particular [math]\displaystyle{ \mathcal{H}_1 }[/math] that it cannot improve (or more generally, a set of similar [math]\displaystyle{ \mathcal{H}_1 }[/math]'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

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Iterated merging can be viewed as a form of simulated annealing. The set [math]\displaystyle{ \mathcal{S} }[/math] initially contains at least two admissible [math]\displaystyle{ k_0 }[/math]-tuples (typically many more), and as the algorithm proceeds the set [math]\displaystyle{ \mathcal{S} }[/math] converges toward [math]\displaystyle{ \mathcal{H}_1 }[/math] and the number of admissible [math]\displaystyle{ k_0 }[/math]-tuples it contains declines. One can regard the cardinality of the difference between [math]\displaystyle{ \mathcal{S} }[/math] and [math]\displaystyle{ \mathcal{H}_1 }[/math] as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given [math]\displaystyle{ \mathcal{S} }[/math]).

Local optimizations

Let [math]\displaystyle{ \mathcal H = \{h_1,\ldots, h_{k_0}\} }[/math] be an admissible [math]\displaystyle{ k_0 }[/math]-tuple with endpoints [math]\displaystyle{ h_1 }[/math] and [math]\displaystyle{ h_{k_0} }[/math], and let [math]\displaystyle{ \mathcal I }[/math] be the interval [math]\displaystyle{ [h_1,h_{k_0}] }[/math]. If there exists an integer [math]\displaystyle{ h\in\mathcal I }[/math] such that removing one of [math]\displaystyle{ \mathcal H }[/math]'s endpoints and inserting [math]\displaystyle{ h }[/math] yields an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal H' }[/math], then call [math]\displaystyle{ \mathcal H }[/math] contractible, and if not, say that [math]\displaystyle{ \mathcal H }[/math] non-contractible. Note that [math]\displaystyle{ \mathcal H' }[/math] necessarily has smaller diameter than [math]\displaystyle{ \mathcal H }[/math]. Any of the sieving methods described above may produce admissible [math]\displaystyle{ k_0 }[/math]-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing [math]\displaystyle{ \mathcal H }[/math] by [math]\displaystyle{ \mathcal H' }[/math] if this test succeeds.

We can also shift [math]\displaystyle{ \mathcal H }[/math] to the left by removing its right end point [math]\displaystyle{ h_{k_0} }[/math] and replacing it with the greatest integer [math]\displaystyle{ h_0 \lt h_1 }[/math] that yields an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal H' }[/math], and we can similarly shift [math]\displaystyle{ \mathcal H }[/math] to the right. The diameter of [math]\displaystyle{ \mathcal H' }[/math] need not be less than [math]\displaystyle{ \mathcal H }[/math], but if it is, it provides a useful replacement. More generally, by shifting [math]\displaystyle{ \mathcal H }[/math] repeatedly we can produce a sequence of admissible [math]\displaystyle{ k_0 }[/math]-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted [math]\displaystyle{ \mathcal H' }[/math] with smaller diameter than [math]\displaystyle{ \mathcal H }[/math].

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A more sophisticated local optimization involves a process of ``adjustment" proposed by Savitt. Let [math]\displaystyle{ \mathcal H }[/math] be an admissible [math]\displaystyle{ k_0 }[/math]-tuple. For a prime [math]\displaystyle{ p }[/math] and an integer [math]\displaystyle{ a }[/math], let [math]\displaystyle{ [a;p] }[/math] denote the residue class [math]\displaystyle{ a\bmod p }[/math], i.e. the set of integers [math]\displaystyle{ \{ x : x = a \bmod p\} }[/math]. Call [math]\displaystyle{ [a;p] }[/math] occupied if it contains an element of [math]\displaystyle{ \mathcal H }[/math].

Suppose that [math]\displaystyle{ [a;p] }[/math] and [math]\displaystyle{ [b;q] }[/math] are occupied residue classes, for some distinct primes [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math], and that [math]\displaystyle{ [a';p] }[/math] and [math]\displaystyle{ [b';q] }[/math] are unoccupied. Let [math]\displaystyle{ \mathcal U }[/math] be the intersection of [math]\displaystyle{ \mathcal H }[/math] with [math]\displaystyle{ [a;p] \cup [b;q] }[/math], and let [math]\displaystyle{ \mathcal V }[/math] be a subset of the integers that lie in the intersection of the interval [math]\displaystyle{ I }[/math] containing [math]\displaystyle{ H }[/math] and the set [math]\displaystyle{ [a';p] \cup [b';q] }[/math] such that the set [math]\displaystyle{ \mathcal H' }[/math] formed by removing the elements of [math]\displaystyle{ \mathcal U }[/math] from [math]\displaystyle{ \mathcal H }[/math] and adding the elements of [math]\displaystyle{ \mathcal V }[/math] is admissible. A necessary (and often sufficient) condition for and integer [math]\displaystyle{ v }[/math] to lie in [math]\displaystyle{ \mathcal V }[/math] is that [math]\displaystyle{ v }[/math] must not lie in a residue class [math]\displaystyle{ [c;r] }[/math] that is the unique unoccupied residue class modulo [math]\displaystyle{ r }[/math] for any prime [math]\displaystyle{ r }[/math] other than [math]\displaystyle{ p }[/math] or [math]\displaystyle{ q }[/math].

The admissible set [math]\displaystyle{ \mathcal H' }[/math] lies in the interval [math]\displaystyle{ \mathcal I }[/math] containing [math]\displaystyle{ \mathcal H }[/math], so its diameter is no greater than that of [math]\displaystyle{ \mathcal H }[/math], however its cardinality may differ. If it happens that [math]\displaystyle{ \mathcal H' }[/math] contains more elements than [math]\displaystyle{ \mathcal H }[/math], then by eliminating points at either end of [math]\displaystyle{ \mathcal H' }[/math] we obtain an admissible [math]\displaystyle{ k_0 }[/math]-tuple that is narrower than [math]\displaystyle{ \mathcal H }[/math] and may ``adjust" [math]\displaystyle{ \mathcal H }[/math] by replacing it with [math]\displaystyle{ \mathcal H' }[/math]. The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible [math]\displaystyle{ k_0 }[/math]-tuples.

Lower bounds

There is a substantial amount of literature on bounding the quantity [math]\displaystyle{ \pi(x+y)-\pi(x) }[/math], the number of primes in a shifted interval [math]\displaystyle{ [x+1,x+y] }[/math], where [math]\displaystyle{ x,y }[/math] are natural numbers. As a general rule, whenever a bound of the form

[math]\displaystyle{ \pi(x+y) - \pi(x) \leq F(y) }[/math] (*)

is established for some function [math]\displaystyle{ F(y) }[/math] of [math]\displaystyle{ y }[/math], the method of proof also gives a bound of the form

[math]\displaystyle{ k_0 \leq F( H(k_0)+1 ). }[/math] (**)

Indeed, if one assumes the prime tuples conjecture, any admissible [math]\displaystyle{ k_0 }[/math]-tuple of diameter [math]\displaystyle{ H }[/math] can be translated into an interval of the form [math]\displaystyle{ [x+1,x+H+1] }[/math] for some [math]\displaystyle{ x }[/math]. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for [math]\displaystyle{ x\gt y }[/math], the set of primes between [math]\displaystyle{ x+1 }[/math] and [math]\displaystyle{ x+y }[/math] is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

Brun-Titchmarsh inequality

The Brun-Titchmarsh theorem gives

[math]\displaystyle{ \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y} }[/math]

which then gives the lower bound

[math]\displaystyle{ H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0 }[/math].

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

[math]\displaystyle{ k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}. }[/math]

First Montgomery-Vaughan large sieve inequality

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

[math]\displaystyle{ k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2 }[/math]

for any [math]\displaystyle{ Q \gt 1 }[/math], which is a parameter that one can optimise over (the optimal value is comparable to [math]\displaystyle{ H(k_0)^{1/2} }[/math]).

Second Montgomery-Vaughan large sieve inequality

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

[math]\displaystyle{ k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1} }[/math]

for any [math]\displaystyle{ z \gt 1 }[/math], which is a parameter similar to [math]\displaystyle{ Q }[/math] in the previous inequality, and [math]\displaystyle{ c }[/math] is an absolute constant. In the original paper of Montgomery and Vaughan, [math]\displaystyle{ c }[/math] was taken to be [math]\displaystyle{ 3/2 }[/math]; this was then reduced to [math]\displaystyle{ \sqrt{22}/\pi }[/math] [B1995, p.162] and then to [math]\displaystyle{ 3.2/\pi }[/math] [M1978]. It is conjectured that [math]\displaystyle{ c }[/math] can be taken to in fact be [math]\displaystyle{ 1 }[/math].

Benchmarks

Efforts to fill in the blank fields in this table are very welcome.

[math]\displaystyle{ k_0 }[/math]	3,500,000	181,000	34,429	26,024	23,283	22,949	10,719	6,329	5,000
Upper bounds
First [math]\displaystyle{ k_0 }[/math] primes past [math]\displaystyle{ k_0 }[/math]	59,874,594	2,530,338	420,878	310,134	275,082	270,698	117,714		50,840
Zhang sieve	59,093,364	2,486,370	411,932	303,558	268,536	264,414	114,806	64,176	49,578
Hensley-Richards sieve	57,554,086	2,422,558	402,790	297,454	262,794	258,780	112,868		48,634
Asymmetric Hensley-Richards		2,418,054	401,700	296,154	262,286	258,302	112,562		48,484
Shifted Schinzel sieve		2,413,228	400,512	295,162	262,206	258,000	112,440		48,726
Greedy-greedy sieve		2,326,476	388,076	286,308	253,968	249,992	108,694		46,968
Best known tuple	57,554,086	2,326,476	386,382	285,210	252,804	248,898	108,450	60,744	46,810
Predictions
[math]\displaystyle{ k_0 \log k_0 + k_0 }[/math]	56,238,957	2,372,232	394,096	290,604	257,405	253,381	110,119	61,727	47,586
Lower bounds
MV with [math]\displaystyle{ c=1 }[/math] (conjectural)	32,503,908	1,395,694	234,872	173,420	153,691	151,298	66,314	37,274	28,781
MV with [math]\displaystyle{ c=3.2/\pi }[/math]	32,469,985	1,393,869	234,529	173,140	153,447	151,056	66,211	37,207	28,737
MV with [math]\displaystyle{ c=\sqrt{22}/\pi }[/math]	31,765,216	1,357,096	227,078	167,860	148,719	146,393	63,917	35,903	27,708
Second Montgomery-Vaughan	31,756,667	1,356,644	226,987	167,793	148,656	146,338	63,886	35,887	27,696
Brun-Titchmarsh	30,137,225	1,272,083	211,046	155,555	137,756	135,599	58,863	32,916	25,351
First Montgomery-Vaughan	28,080,007	1,184,954	196,729 196,719 197,096	145,711 145,461	128,971	126,931	55,149 55,178	30,982	24,012 24,037

[math]\displaystyle{ k_0 }[/math]	4,000	3,405	3,000	2,000	1,000	672
Upper bounds
First [math]\displaystyle{ k_0 }[/math] primes past [math]\displaystyle{ k_0 }[/math]	39,660		28,972	18,386	8,424	5,406
Zhang sieve	38,596		28,008	17,766	8,212	5,216
Hensley-Richards sieve	38,498		27,806	17,726	8,258	5,314
Asymmetric Hensley-Richards	37,932		27,638	17,676	8,168	5,220
Shifted Schinzel sieve	38,168		27,632	17,616	8,160	5,196
Greedy-greedy sieve	36,756		26,754	17,054	7,854	5,030
Best known tuple	36,636*	30,600	26,606	16,984*	7,806*	5,010*
Engelsma data	36,622	30,606	26,622	16,978	7,802	4,998
Predictions
[math]\displaystyle{ k_0 \log k_0 + k_0 }[/math]	37,176		27,019	17,202	7,907	5,046
Lower bounds
MV with [math]\displaystyle{ c=1 }[/math] (conjectural)	22,564	18,898	16,456	10,500	4,858	3,124
MV with [math]\displaystyle{ c=3.2/\pi }[/math]	22,523	18,866	16,428	10,480	4,847	3,118
MV with [math]\displaystyle{ c=\sqrt{22}/\pi }[/math]	21,701	18,153	15,758	10,061	4,648	2,979
Second Montgomery-Vaughan	21,690	18,143	15,751	10,056	4,645	2,977
Brun-Titchmarsh	19,785	16,536	14,358	9,118	4,167	2,648
First Montgomery-Vaughan	18,768 18,859	15,783	13,696	8,448 8,615	3,959	2,558

* indicates that the widths listed are the best known tuples that have been found by the methods that gave the entries for larger values of [math]\displaystyle{ k_0 }[/math], but are not as narrow as the literally best known tuples (due to Engelsma).

For the Zhang tuples the optimal [math]\displaystyle{ m \lt \pi(10^{10}) }[/math] that produced an admissible [math]\displaystyle{ k_0 }[/math]-tuple was used. This is not always the least [math]\displaystyle{ m }[/math] that produces an admissible [math]\displaystyle{ k_0 }[/math]-tuple; for [math]\displaystyle{ k_0= }[/math]22,949, for example, the minimal [math]\displaystyle{ m= }[/math]586 yields an admissible [math]\displaystyle{ k_0 }[/math]-tuple of diameter of 264,460, but [math]\displaystyle{ m= }[/math]599 yields a narrower admissible [math]\displaystyle{ k_0 }[/math]-tuple with the listed diameter of 264,414. A list of table entries for which this occurs can be found here (and also for [math]\displaystyle{ k_0 }[/math]=6,329).

The shifted Schinzel tuples were generated with [math]\displaystyle{ y=2 }[/math] using an optimally chosen interval contained in [math]\displaystyle{ [-k_0\log k_0, k_0\log k_0] }[/math] (the interval is not in every case guaranteed to be optimal, particularly for larger values of [math]\displaystyle{ k_0 }[/math], but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in [math]\displaystyle{ [-k_0\log k_0, k_0\log k_0] }[/math] was selected on this basis). As noted by Castryck, breaking ties upward may produce better results in some cases.