MATHEMATICS OF COMPUTATION Volume 84, Number 291, January 2015, Pages 309–337 S 0025-5718(2014)02837-8 Article electronically published on April 8, 2014
ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS WITH SMALL MULTIPLE SEEDS
ZHENXIANG ZHANG Dedicated to Guy Robin on the occasion of his 75th birthday
Abstract. For a positive even integer L,letP(L)denotethesetofprimes p for which p − 1 divides L but p does not divide L,letC(L)denotetheset of Carmichael numbers n where n is composed entirely of primes in P(L)and where L divides n − 1, and let W(L) ⊆C(L) denote the subset of Williams numbers, which have the additional property that p +1 | n +1 foreach prime p | n. We study |C(L)| and |W(L)| for certain integers L. We describe procedures for generating integers L that have more even divisors than any smaller positive integer, and we obtain certain numerical evidence to support − |C | s(1+o(1)) |W | s1/2 o(1) the conjectures that log2 (L) =2 and log2 (L) =2 when such an “even-divisor optimal” integer L has s different prime factors. For example, we determine that |C(735134400)| > 2 · 10111.Last,usinga heuristic argument, we estimate that more than 224 Williams numbers may be manufactured from a particular set of 1029 primes, although we do not construct any explicit examples, and we describe the difficulties involved in doing so.
1. Introduction
Let bi be the ith prime and let s ei (1.1) L = bi i=1
with s ≥ 2, ei ≥ 1fori ∈{1,s} and ei ≥ 0for1
Received by the editor June 26, 2012 and, in revised form, January 6, 2013, April 8, 2013 and May 1, 2013. 2010 Mathematics Subject Classification. Primary 11Y16, 11Y35; Secondary 11Y11. Key words and phrases. Carmichael numbers (with small multiple seeds), Korselt’s criterion, Erd˝os’s construction of Carmichael numbers, even-divisors-optimal numbers (EDONs), Williams numbers, algorithms. The author was supported by the NSF of China, Grant 10071001.
c 2014 American Mathematical Society Reverts to public domain 28 years from publication 309
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and (1.5) C(L)={n ∈N(L):L | n − 1,n− 1 = L if L + 1 is prime}. If L is square-free, we say that these sets are generated by the (square-free) { } { kj ≤ (prime) seeds bj , otherwise these sets are generated by multiple seeds bj : kj ej }. By Korselt’s criterion [13] (see also [7, Section 3.4.2]), every number n ∈C(L) is Carmichael [5]. Let As be the L in (1.1) with each ei = 1, i.e., As is the product of the first s primes. Erd˝os [11] argued that, for any ε>0 and all sufficiently large x (depending 1−ε on the choice of ε), the set C(As)containsmorethanx Carmichael numbers ε/4 ≤ x,wheres is the largest number such that bs < ln x . In short, Erd˝os [11] made the following Conjecture 1. Conjecture 1 (Erd˝os). There are x1−o(1) Carmichael numbers up to x. Based on Erd˝os’s original heuristic [11], though with certain modification, Alford, Granville, and Pomerance [2] proved that there are more than x2/7 Carmichael numbers up to x,oncex is sufficiently large. In [21] we gave some reasons and numerical evidence to support the following Conjecture 2. Conjecture 2. We have 2s(1−o(1)) (1.6) |C(As)| =2 .
In [21] we described a procedure for finding |C(As)| for small s and tabulated |C(As)| for 3 ≤ s ≤ 10. In particular, we have 29·(1−0.363...) |C(A9)| = 8281366855879527 = 2 and |C(A10)| = 21823464288660480291170614377509316 (33 digits) 10·(1−0.316...) =22 . 1.2. Williams numbers with small seeds. Williams [19] asked whether there are any Carmichael numbers n with an odd number of prime divisors and the additional property that for each prime p | n, (1.7) p +1| n +1. Echi [9] omitted the phrase “with an odd number of prime divisors” to relax the Williams question. Echi [9] then said: “This is a long standing open question; and it is possible that there is no such number.” So, he extended the problem of Williams: he called a square-free composite n an a-Williams number if p − a | n − a and p + a | n + a for all primes p dividing n,wherea is a given nonzero integer. Bouall`egue, et al. [3] found some a-Williams numbers n<108, with a ≈ n/3. In contrast to Echi, we believe that there should be infinitely many Carmichael numbers satisfying (1.7) (see §6 for the reasons.) We call such Carmichael numbers Williams numbers (which are 1-Williams numbers by Echi), though so far not a single Williams number has been found; see also [20, Remark 5.2].
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Let L be as defined in (1.1). Let (1) ej (1.8) L =2 bj 2≤j≤s 2|j
and (2) ej (1.9) L = bj . 1≤j≤s 2j Then L(1)L(2) gcd(L(1),L(2))=2, and lcm(L(1),L(2))= = L. 2 Let (1.10) Q(L)= prime q : q − 1 ∈D(L(1)),q+1∈D(L(2)),q L , N (L)= n>1:n is square-free and composed entirely (1.11) of the primes in Q(L) , (1.12) C(L)= n ∈N(L):L(1) | n − 1 ,
and W(L)= n ∈C(L):L(2) | n +1 , (1.13) = n ∈N(L):L(1) | n − 1,L(2) | n +1 . Then every number n ∈W(L) is Williams. In this paper, we evaluate both |C(L)| and |W(L)| for general (nonsquare-free) numbers L. For stating our main results, we need some definitions. We state our definitions and main results in the following section.
2. Definitions and main results Definition 2.1. A positive integer L is called an even-divisors-optimal number (EDON )if|D(L)| < |D(L)| for any positive integer L less than L,whereD(L)is as defined by (1.2). Definition 2.2. If
(2.1) e1 >e2 ≥ ...≥ es ≥ 1 in (1.1), we say that the number L is regular,otherwiseL is irregular. Definition 2.3. A positive integer L is called a Carmichael conductor,iftheset C(L) is not empty. Analogously, a positive integer L is called a Williams conductor, if the set W(L)isnotempty. The smallest square-free Carmichael conductor is 2310 = 2 · 3 · 5 · 7 · 11, while the smallest nonsquare-free Carmichael conductor is 36 = 22 · 32, noting that |P(2310)| =9, |P(36)| =5, |C(2310)| = |C(36)| =2.
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The main purpose of this paper is to give some reasons and certain numerical evidence to support Conjectures 3 and 4, where ω(L) is the number of different prime divisors of L. Conjecture 3. For an EDON L with ω(L)=s, we have |C | s(1+o(1)) (2.2) log2 (L) =2 . Conjecture 4. For an EDON L with ω(L)=s, we have |W | s1/2−o(1) (2.3) log2 (L) =2 . Our reasons for making Conjectures 3 and 4 are mainly based on the heuristics of Erd˝os, Alford, Granville, and Pomerance concerning Erd˝os’s construction of Carmichael numbers and based on Theorem 1. Theorem 1. EDONs must be regular. If L is an EDON with ω(L)=s,thenwe have 2s (2.4) L<2bs+1 and (2.5) |D(L)| =2s(1+o(1)). In Section 3, we prove Theorem 1. In Section 4 we describe an algorithm, based on Theorem 1, to generate EDONs and tabulate some of them with relative values which are necessary for next sections. In Section 5, we give numerical evidence to support Conjecture 3. By an adaptation of the procedure described in [21, Section 2] for computing |C(As)| to compute |C(L)| for a general nonsquare-free L, we find that |C(L)| =243222260471570859335055803087306258056047773279626666779255 9185137636762273565844572484881859106918537019969349 (112 digits) which seems currently to be the largest known cardinality of a set of Carmichael numbers, where L = 735134400 = 26 · 33 · 52 · 7 · 11 · 13 · 17 is the largest EDON less than 109 (see Remark 5.1 for details.) In Section 6, we describe procedures for obtaining numerical evidence of |Q(L)| to support Conjec- ture 4. We exhibit such EDONs L that could be Williams conductors, though we have not actually found any Williams numbers. In Section 7, we tabulate a table of 1029 primes, and believe that there could be at least 224 Williams numbers made up from them, based on an equi-distribution assumption. We state the difficulties for explicitly finding Williams numbers (Remark 7.1), and predict what a Williams number looks like (Remarks 7.2 and 7.4). Remark 2.1. A highly composite number (HCN) is a positive integer with more divisors (both even and odd) than any positive integer smaller than itself. HCNs were first defined by S. Ramanujan [16], and then were studied by Alaoglu and Erd˝os [1], Erd˝os [10], Nicolas [14] and Robin [17]. The concept of EDONs in this paper seems to be new, which emphasizes even divisors of a number L, so that there would be more primes in P(L)andQ(L), and thus there would be more Carmichael numbers in C(L) (Remark 5.2) and more Williams numbers in W(L) (Remark 7.5), though the smallest Carmichael conductor 36 is an HCN, not an EDON.
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3. Proof of Theorem 1 By (1.1) and (1.2), it is easy to see that s (3.1) |D(L)| = e1 (ei +1). i=2 In the rest of this section, let L (given by (1.1)) be an EDON. To prove Theo- rem 1, we need eight lemmas. Lemma 3.1. EDONs must be regular. Proof. The lemma follows by (3.1) and Definitions 2.1 and 2.2. Lemma 3.2. We have ≤ ≤ log2 bs +1 e1 2 log2 bs+1 +1. ≤ Proof. Suppose e1 log2 bs . Then
e1 (3.2) 2 (3.4) ej + uj +1≥ 2(ej +1). Let L · buj L = j . bs Then L License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 314 ZHENXIANG ZHANG Lemma 3.4. If ej is odd for some j with 2 ≤ j (ej +1)/2 ≥ Case 1(ej is odd). Suppose bj >bs+1 or ej 2 logbj bs+1 +1. Let L · b L = s+1 . (ej +1)/2 bj Then L Lemma 3.5. We have 9 s 1/2 max{j : e ≥ 2} > , j 5 ln s once s is sufficiently large. 1/2 Proof. If bj License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 315 and since (3.6) bs ≥ s(ln s +lnlns − 1) for s ≥ 2 and (3.7) bs ≤ s(ln s +lnlns − 0.9484) for s ≥ 39017, a simple computation shows that 9 s 1/2 π(b1/2) > , s 5 ln s once s is sufficiently large. The lemma follows. Lemma 3.6. We have 1/2 |D(L)| > 2s · (3/2)(s/ ln s) , once s is sufficiently large. Proof. By (3.1) and Lemma 3.5, 1/2 1/2 |D(L)| > 3(s/ ln s) · 2s−(s/ ln s) . The lemma follows. Lemma 3.7. We have 13/16 #{j : ej ≥ 2} License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 316 ZHENXIANG ZHANG Proof. By Lemmas 3.2 and 3.4, e1 < 2log2 bs+1 +1, and ej < 2logbj bs+1 < 2log2 bs+1 for j ≥ 2. Then s ej 2s L = bj < 2bs+1 j=1 and |D | s13/16 · s−s13/16 (L) = e1 (ej +1)< (2 log2 bs+1 +1) 2 2≤j≤s by Lemma 3.7. The lemma follows. Theorem 1 follows by Lemmas 3.1, 3.6 and 3.8. 4. An algorithm for generating EDONs Given s ≥ 1, let (4.1) Ls =max{L : L is an EDON with ω(L)=s}. Then 8 = L1 (4.2) Ls = {n1 be the sequence of the first t = t(s) consecutive EDONs with nt = Ls,whereni is the i-th EDON with n1 =2.Then{2, 4, 8} = L1 ⊂L2 ⊂ ··· and 3 = t(1) < t(2) < ···. In this section, we will give an algorithm (Procedure 4.3) to determine Ls for any s>1. Procedure 4.2 is a subroutine of Procedure 4.3; Procedure 4.1 is a subroutine of Procedure 4.2. Let s ≥ 2 and let n be a positive integer with ω(n)=s.If s ei n = bi i=1 satisfying Lemmas 3.1—3.4, then we call n an s-candidate.Let L = {n1 be a dynamic sequence with t = |L| and mi = |D(ni)|, initialized with L = {2, 4, 8} and t =3.Anelementnk of L is called an r-survived,ifω(nk)=r. Let n be an s-candidate with m = |D(n)|. Procedure 4.1 mainly does one of the following four tasks for n: (1) gives up this n if there is some nk ∈Lsuch that nk License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 317 begin if i = j then // doing (2) or (3) Begin j ← j +1; If m = mj Then nj ← n // doing (2) Else // doing (3) begin t ← t +1;i ← t; repeat mi ← mi−1; ni ← ni−1; i ← i − 1 until i = j; mi ← m; ni ← n end End else // doing (4) Begin j ← j +1;mj ← m; nj ← n; If m = mi+1 Then i ← i +1; repeat j ← j +1;i ← i +1;mj ← mi; nj ← ni until i = t; t ← j End end End. PROCEDURE 4.2. Handle-one-s; // subroutine of Procedure 4.3, generating all s-candidates Begin For each e1 satisfying Lemma 3.2 Do begin for each ej (2 ≤ j ≤ s) satisfying Lemmas 3.3 and 3.4 do Begin If (e1,e 2,...,es) satisfies (2.1) Then ← s ei begin n i=1 bi ;//nown is an s-candidate ← s |D | m e1 i=2(ei + 1); // with m = (n) Handle-one-n end End end End. L PROCEDURE 4.3. Determining the sequence s0 ; // main procedure ≥ L //input a positive integer s0 2, determine the sequence s0 BEGIN n1 ← 2; n2 ← 4; n3 ← 8; s ← 1; t ← 3; // initializing L m1 ← 1; m2 ← 2; m3 ← 3; Repeat s ← s + 1; Handle-one-s Until s = s0; Repeat s ← s + 1; Handle-one-s Until the smallest s-survived > the largest s0-survived; ← Ls0 the largest s0-survived; ← t0 k if nk = Ls0 ; L ← L s0 the subsequence of the first t0 elements of END. A Delphi-Pascal program (with multi-precision package partially written in As- sembly language) ran several minutes on my PC (Pentium Dual E2180/2.0GHz with 1.99 GB memory) to get the sequence L136 with t0 = |L136| = 7322, among which there are 30 EDONs L with ω(L) = 137, no EDON L with ω(L) > 137. The first 100 EDONs are tabulated in the supplementary Table 8. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 318 ZHENXIANG ZHANG In the supplementary Table 9 we tabulate some values relative to Ls for 3 ≤ s ≤ 136, where ⎧ ⎪ms,1 = log Ls + 1 (the number of decimal digits of Ls), ⎨⎪ 10 m = log h + 1 (the number of decimal digits of h ), (4.3) s,2 10 s,0 s,0 ⎪ln(L )=s1+αs , ⎩⎪ s βs =log2(hs,0), with s(1+εs,0) (4.4) hs,0 = |D(Ls)| =2 . For saving typeset space in the supplementary Tables 8 and 9, when consecutive ei arethesame,wesimplywrite“x:y” which means that there are y occurrences of exponent ei equal to x, e.g., “5 2:2 1:3” stands for one 5, two 2s and three 1s. In Tables 1, 3, 5, 6, and 7, and in the supplementary Table 9, we omit “ ...”in decimals, e.g., we simply write “0.805993” instead of “0.805993 ...”. Remark 4.1. We see from the supplementary Table 9 that, for EDON L with ω(L)=s, (4.5) ln L for s ≥ 93, and that values of εs,0 tend to 0 as stated in (2.5) of Theorem 1. 5. Counting and estimating the count of Carmichael numbers with small multiple seeds Let L be an EDON with ω(L)=s.LetD(L), P(L), N (L), and C(L)beas defined in Section 1. Analogously, by Erd˝os’s argument [11] that members of the set N (As)areroughly equi-distributed mod As, members of the set N (L)areroughly equi-distributed mod L.Thuswewouldhave (5.1) |C(L)|≈a(L) where |N (L)| 2|P(L)| − 1 2|P(L)| − 1 (5.2) a(L)= = > . φ(L) φ(L) L Since the probability of a number ≤ m to be prime is greater than 1/ ln m,itis reasonable to believe that (the second inequality is trivial) |D(L)| (5.3) < |P(L)| < |D(L)|. ln L By (5.3) and Theorem 1, we have |D(L)| 2s(1+εs,0) (5.4) |P(L)| > > =⇒|P(L)| =2s(1+εs,1). ln L 2s ln bs+1 +ln2 Then by (5.2), (5.4), and Theorem 1, we have |P(L)| 2s(1+εs,1) 2 − 1 2 − 1 ⇒ s(1+εs,2) (5.5) a(L) > > 2s = log2 a(L)=2 . L 2bs+1 Thus by (5.1) and (5.5), we have Conjecture 3. Let Ls be as defined in (4.1) and let hs,0 = |D(Ls)| be as defined by (4.4) . To find exact values of |P(Ls)| to support (5.4) and (5.5), we should check whether d+1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 319 is prime for each d ∈D(Ls). We use Procedure 5.2 to do this task. To determine all d ∈D(Ls), Procedure 5.2 uses only hs,0/e1 −1 multiplications (multiply a multi- precision integer by a single-precision integer) and hs,0 −hs,0/e1 additions (double a multi-precision integer). If we define an operation as one multiplication of a multi- precision integer by a single-precision integer, or doubling a multi-precision integer, then Procedure 5.2 needs only hs,0 − 1 operations to determine hs,0 elements d ∈ D(Ls). Procedure 5.1 is a subroutine of Procedure 5.2 and calls itself recursively. For checking whether d +1 ∈P(L)ford ∈D(L), we use Lucas’s converse of Fermat’s little theorem (see [7, Theorem 4.1.1] or [18, Theorem 9.18]) or even better, one may use the Brillhart et al.’s relaxed version [4] of Lucas’s theorem. PROCEDURE 5.1. next(j, u0); // a subroutine of Procedure 5.2 and calls itself recursively // checking whether d + 1 is prime for each d (∈D(Ls)) having odd seeds // parameters j and u0: single precision integers (d already has u0 odd seeds // counted with multiplicity; consider whether or not bj | d) // local variables u, k, and t: single-precision integers // main variables (defined in Procedure 5.2) h0 and h1: single-precision integers; // d: multi-precision integer; n: an array of multi-precision integers // initialize in Procedure 5.2: next(s, 0) BEGIN If j>2 Then next(j − 1,u0); //case 1: bj d u ← u0; //case 2: bj | d For k := 1 To ej Do ← ← · k | begin u u +1; nu nu−1 bj ; //case 2.1: 2bj d if j>2 then next(j − 1,u); d ← nu; h0 ← h0 +1; if (d>bs) and (d + 1 is prime) then h1 ← h1 +1; − t+1 k | for t := 1 to e1 1 do //case 2.2: 2 bj d Begin d ← d + d; h0 ← h0 +1; If (d>bs) And (d + 1 is prime) Then h1 ← h1 +1 End end END. PROCEDURE 5.2. Determining |P(Ls)|;// input Ls; output |P(Ls)| // variables h0,h1, and r: single-precision integers // d: multi-precision integer; n: an array of multi-precision integers BEGIN n0 ← 2; d ← n0; h0 ← 1; h1 ← 0; //initializing r+1 For r := 1 To e1 − 1 Do //checking whether d + 1 is prime for d =2 begin d ← d + d; h0 ← h0 +1; if (d>bs) and (d + 1 is prime) then h1 ← h1 +1 end; next(s, 0); //checking whether d + 1 is prime for each d having odd seeds If h0 = hs,0 Then begin output message “error!”; exit end; |P(Ls)|←h1 END. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 320 ZHENXIANG ZHANG A Delphi-Pascal program ran about several hours on my PC to find |D(Ls)|, |P(Ls)| and relative values for 3 ≤ s ≤ 20 tabulated in Table 1. Since |D(Ls)| in- creases exponentially on s, the running time of the program increases exponentially on s. It is not difficult to adapt Procedure 1 in [21, Section 2] for computing |C(As)| to compute |C(L)| for a general nonsquare-free L<108. A Delphi-Pascal program (with multi-precision package partially written in Assembly language) ran in less than an hour on my PC to get the exact values of |C(Ls)| for 3 ≤ s ≤ 6 tabulated in Table 2. In Table 3, we give comparisons of |C(Ls)| and a(Ls)for3≤ s ≤ 6, where C s(1+εs,2) (5.6) log2 (Ls)=2 . From Table 3 we see that numerical evidence for s ≥ 4 supports (5.1), i.e., numerical evidence for s ≥ 4 supports Erd˝os’s argument [11] that members of the set Ns are roughly equi-distributed mod Ls. Thus we should believe that a(Ls)servesas|C(Ls)| ≥ and εs,2 serves as εs,2 for all s 4. So, Conjecture 3 is supported. Table 1. |D(Ls)|, |P(Ls)| and relative values for 3 ≤ s ≤ 20 |D | |P | s (Ls) (Ls) εs,1 log2 φ(Ls) log2 a(Ls) εs,2 3 30 13 0.233 8.584 4.415 −0.285 4 108 48 0.396 14.491 33.508 0.266 5 224 88 0.291 18.076 69.923 0.225 6 672 241 0.318 23.983 217.016 0.293 7 1680 549 0.300 29.568 519.431 0.288 8 5040 1452 0.312 36.545 1415.454 0.308 9 12960 3279 0.297 43.005 3235.994 0.295 10 25920 6071 0.256 47.812 6023.187 0.255 15 1492992 237577 0.190 79.171 237497.828 0.190 20 99532800 11749381 0.174 115.570 11749265.429 0.174 Table 2. |C(Ls)| for 3 ≤ s ≤ 6 s |C(Ls)| 3 26 4 12216662176 5 1119375758885937207949 6 213016862555264314426998637735311846600581943524068771133901041653 Table 3. Comparisons of |C(Ls)| and a(Ls)for3≤ s ≤ 6 |C | s log2 (Ls) εs,2 log2 a(Ls) εs,2 3 4.700439 −0.255734758 4.415037 −0.285858102 4 33.508131 0.266609829 33.508146 0.266609999 5 69.923184 0.225539796 69.923184 0.225539796 6 217.016293 0.293609925 217.016293 0.293609925 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 321 Remark 5.1. From the supplementary Table 8 we see that there are several EDONs L with the same ω(L), e.g., there are 14 EDONs L with ω(L)=7.Let L = 735134400 = 26 · 33 · 52 · 7 · 11 · 13 · 17 be the 10th EDON with ω(L) = 7 (or the 63rd EDON of all). For computing |C(L)|, we use the strategies and tactics described in [21, Remarks 2.2 and 2.3] so that the interim data can fit into 1.99 GB of memory. A Delphi-Pascal program ran about 350 hours on my PC to get the exact values of |C(L)| stated in Section 2, with |C | 7·(1+0.288...) log2 (L) =2 . Since the largest EDON with ω(L)=7is 7 4 2 L7 = 4410806400 = 2 · 3 · 5 · 7 · 11 · 13 · 17, with φ(L7) = 796262400 six times larger than φ(735134400) = 132710400, much more time would be used to get |C(L7)|, so we did not run the program. However, we believe that Conjecture 3 is valid for any EDON L with ω(L)=s once s is sufficiently large. Remark 5.2. Alford (see [12]) took 6 3 2 2 LAlf = 23284800 = 2 · 3 · 5 · 7 · 11, determined 155 primes p for which p − 1 divides L, and then established that there 128 are at least 2 − 1 Carmichael numbers made up from them. However, LAlf is not an EDON — there are three EDONs L |D(L)|≥|D(LAlf)| = 432, |P(L)| > |P(LAlf)| = 155, and |C(L)| > |C(LAlf)| = 9439468867102976341813032284832753452367 (40 digits); |C | |C | see Table 4, where m = log10 (L) + 1 (decimal digits of (L) ). The three Ls are explicit examples explaining Remark 2.1. See also [21, Remark 1.1]. Table 4. Three EDONs L L |D(L)| |P(L)| |C(L)| m factorization of L 14414400 432 160 528610 ...557174 42 26 · 32 · 52 · 7 · 11 · 13 17297280 448 161 881017 ...013067 42 27 · 33 · 5 · 7 · 11 · 13 21621600 480 186 236496 ...715974 50 25 · 33 · 52 · 7 · 11 · 13 6. Estimating the count of Williams numbers with small multiple seeds (1) Let s ≥ 10 and let L = Ls be as defined by (4.1). Then ω(L)=s.LetL , L(2), D(L), Q(L), N (L), C(L), and W(L) be as defined in Section 1. Define the sets (1) (1) (6.1) Hs,1 = D(L )={d ∈D(L):d | L }, (2) (2) (6.2) Hs,2 = D(L )={d ∈D(L):d | L }, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 322 ZHENXIANG ZHANG and (2) (6.3) Hs,3 = {d ∈Hs,1 : d +2∈Hs,2} = {d ∈Hs,1 : d +2| L }. Then Q(L)={prime q>bs : q − 1 ∈Hs,3} (6.4) (1) (2) = {prime q>bs : q − 1 | L ,q+1| L }. Define functions h = |H | for 1 ≤ i ≤ 3, (6.5) s,i s,i hs,4 = |Q(L)|. Then ⎧ ⎨⎪Hs,1 ∩Hs,2 = {2}, (6.6) h · h = h , ⎩⎪ s,1 s,2 s,0 ≈ ≈ 1/2 hs,1 hs,2 hs,0 , where hs,0 = |D(L)| as defined by (4.4). Analogously once again by Erd˝os’s equi-distribution assumption, it is reasonable to believe that members of the set N (L)arebothroughly equi-distributed mod L(1) and roughly equi-distributed mod L.Thuswewouldhave |Q | |C |≈|N (L)| 2 (L) −1 (L) φ(L(1)) = φ(L(1)) , (6.7) |Q | |W |≈|N (L)| 2 (L) −1 (L) φ(L) = φ(L) . Therefore, we would have log |C(L)|≈|Q(L)|−log φ(L(1))=h − log φ(L(1)), (6.8) 2 2 s,4 2 |W |≈|Q |− − log2 (L) (L) log2 φ(L)=hs,4 log2 φ(L), W ∅ which gives us a hint that if hs,4/ log2 φ(L) > 1, then we would have (L) = . So, we must know when we will have hs,4/ log2 φ(L) > 1. First, we use Procedure 6.1 to determine the sets Hs,3 and Q(Ls) for 10 ≤ s ≤ 70. PROCEDURE 6.1. Determining the sets Hs,3 and Q(Ls); // input s ≥ 10, suppose L = Ls is known from the supplementary Table 9 (defined //by (4.1) and computed by Procedure 4.3); determine the sets Hs,3 and Q(Ls) BEGIN Compute L(1) and L(2) by (1.8) and (1.9); H3 ←∅; H4 ←∅; For each d ∈D(L(1)) Do (2) begin if L mod d +2=0then put d into H3 end; Sort H3; For each d (>bs) ∈H3 Do begin if d + 1 is prime then put d +1intoH4 end; Hs,3 ←H3; Q(Ls) ←H4; hs,3 ←|H3|; hs,4 ←|H4| END. To implement the sentence “For each d ∈D(L(1)) Do” we use a recursive subroutine analogous to Procedure 5.1, which needs only hs,1 − 1“operations” (1) to determine hs,1 elements d ∈D(L ). A Delphi-Pascal program (with multi- precision package partially written in Assembly language) ran about 300 hours on License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 323 Table 5. ≤ ≤ (1) hs,3 and hs,4 for 10 s 70 with r =log2 φ(Ls )and r =log2 φ(Ls) s hs,3 hs,4 εs,3 εs,4 r hs,4/r r hs,4/r 10 13 3 0.068253 0.299981 23.7 0.126531 47.8 0.062744 20 43 6 0.064553 0.182978 55.7 0.107703 115.5 0.051916 30 112 18 0.063920 0.080177 94.6 0.190254 193.0 0.093216 40 215 40 0.055035 0.046790 134.9 0.296331 273.9 0.146022 50 394 71 0.050692 0.035684 178.1 0.398484 356.3 0.199260 60 661 105 0.046449 0.034911 221.0 0.474935 440.1 0.238547 70 1152 168 0.045942 0.029141 265.8 0.631917 533.8 0.314703 Table 6. hs,3,k and hs,4,3 for 10 ≤ s ≤ 70 and 0 ≤ k ≤ 3 hs,3,0 hs,3,1 hs,3,2 hs,3,3 s hs,3,0 hs,3,1 hs,3,2 hs,3,3 hs,4,3 hs,3 hs,3 hs,3 hs,3 10 11 84.6% 13 100% 3 20 36 83.7% 39 95.3% 41 95.3% 42 97.6% 6 30 96 85.7% 105 93.7% 108 96.4% 111 99.1% 18 40 192 89.3% 208 96.7% 212 98.6% 213 99.0% 39 50 365 92.6% 379 96.1% 388 98.4% 392 99.4% 71 60 625 94.5% 645 97.5% 655 99.0% 657 99.3% 105 70 1123 97.4% 1143 99.2% 1149 99.7% 1152 100% 168 my PC to get the sets Hs,3 and Q(Ls) for 10 ≤ s ≤ 70 tabulated in Table 5, where εs,3 and εs,4 are such that 1 2 +εs,3 (6.9) log2(hs,3)=s and 1 − 2 εs,4 (6.10) log2(hs,4)=s . From Table 5 we get the following fact: hs,4/ log2 φ(Ls) increases rather fast when s increases, which would be greater than 1 and so W(Ls) would be non- empty when s is large enough. So we should determine the sets Hs,3 and Q(Ls)for larger s. But time needed for running Procedure 6.1 increases exponentially when s increases, we could not afford running Procedure 6.1 for s large enough so that hs,4/ log2 φ(Ls) > 1. However, from the sets Hs,3 and Q(Ls) for 10 ≤ s ≤ 70, we know hs,3,k and hs,4,k for 0 ≤ k ≤ 3, tabulated in Table 6, where k (6.11) Hs,3,k = {d ∈Hs,3 : d k (6.12) Hs,4,k = {q ∈Q(Ls):q From Table 6 we get the following fact: hs,3,3 is very close to hs,3,orinother words, 3 d for almost all d ∈Hs,3. This is reasonably explained as follows. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 324 ZHENXIANG ZHANG Since there are x/2 even numbers not greater than x and the set Hs,2 has hs,2 elements, the probability that a random even number x locates in the set Hs,2 is h s,2 . x/2 Elements d ∈Hs,3 are derived from Hs,1, satisfying d +2∈Hs,2 (2) (i.e., d +2| L ). If we view d + 2 as a random even number for d ∈Hs,1,then the probability that d + 2 locates in the set Hs,2 is h s,2 . (d +2)/2 3 So, if d>hs,2 ·10 , then the probability that d+2 locates in the set Hs,2 is less than 1 ∈H (1) ≈ 1/2 500 . Elements d s,1 spread in a large scope, which can be as large as L L 3 much larger than hs,2 · 10 .So,d + 2 does not locate in the set Hs,2 for almost all 3 larger d of Hs,1 (d>hs,2 · 10 ). For determining hs,3 and hs,4 for s>70, we would rather determine lower bounds of them: hs,3,k and hs,4,k which are very close to hs,3 and hs,4, respectively, where k = 3 or 4. Thus we modified the sentence “ For each d ∈D(L(1)) Do ” of Procedure 6.1 as k (1) “ For each d ( Since the lower bounds h s,3 and h s,4 are very close to the exact values of hs,3 and hs,4 respectively, the derived lower bound ε s,3, upper bounds εs,4 and εs,5 are very very close to the exact values of εs,3, εs,4 and εs,5 respectively, where εs,3 and εs,4 are defined by (6.9) and (6.10) respectively, and εs,5 is such that − |W | s1/2 εs,5 log2 (Ls) =2 . One sees that the data tabulated in Table 7 support Conjecture 4. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 325 Remark 6.1. From Table 7 we see that the set Q(Ls) would “produce” Carmichael numbers for s ≥ 89, and would “produce” Williams numbers for s ≥ 119. Table 7. ≤ ≤ h s,3, h s,4 and relative values for 80 s 131 with (1) r =log2 φ(Ls )andr =log2 φ(Ls) s h s,3 h s,4 ε s,3 εs,4 r h s,4/r r h s,4/r 80 1815 240 0.043567 0.028132 313.9 0.764406 627.0 0.382744 88 2579 342 0.042215 0.024191 349.0 0.979776 699.1 0.489145 89 2701 360 0.042157 0.023439 349.0 1.031343 711.3 0.506058 100 4299 527 0.040849 0.021875 408.2 1.290890 817.3 0.644780 110 6408 752 0.039798 0.019831 457.8 1.642557 913.7 0.823017 118 8721 993 0.039098 0.018278 495.0 2.006024 993.5 0.999409 119 8997 1029 0.038864 0.018052 497.8 2.067027 1004.7 1.024142 120 9338 1063 0.038773 0.017918 507.1 2.095910 1014.1 1.048214 125 11251 1241 0.038398 0.017443 528.9 2.346358 1065.0 1.165226 130 13275 1439 0.037671 0.017105 557.3 2.581816 1112.4 1.293534 131 13797 1497 0.037658 0.016753 560.1 2.672418 1124.7 1.330914 − − s h s,4 r h s,4 r εs,5 89 10.9 100 118.7 110 294.1 118 497.9 119 531.1 24.2 0.180668 120 555.8 48.8 0.139719 125 712.0 175.9 0.083820 130 881.6 326.5 0.063968 131 936.8 372.2 0.060067 7. The set Q(L119) The set Q(L119)={q1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 326 ZHENXIANG ZHANG L =lcm{L(1),L(2)} = 333759 ...800000 (304 digits) ⎛ ⎞ (7.3) 15 119 L(1) · L(2) =214 · 38 · 55 · 75 · 113 · 133 · ⎝ b2⎠ · b = , j j 2 j=7 j=16 (7.4) φ(L(1)) = 720484 ...237248 · 1021 (150 digits), and that (7.5) φ(L) = 286973 ...619648 · 1038 (303 digits). Let n>1 be square-free and composed entirely of the primes in Q(L119). If (7.6) n ≡ 1modL(1), then n is Carmichael. If, in addition, (7.7) n ≡−1modL(2), then n is Williams; or in other words, if n ≡ n0 mod L,thenn is Williams, where n0 = 657637350366 ...522393599999 (303 digits) is the smallest positive solution of the system of congruences (7.6) and (7.7). By (6.8) (see also Table 7), one sees that there would be about h −log φ(L(1)) 531 2 119,4 2 > 2 Carmichael numbers, and about h −log φ(L) 24 2 119,4 2 > 2 Williams numbers, composed entirely of the primes in Q(L119). However, it seems currently beyond our ability to find such a number explicitly; see Remark 7.1. Remark 7.1. It may be reasonable to assume that about 1 out of each φ L(1) members of N (L) is congruent to 1 modulo L(1). So, to find a Carmichael number Q (or a Williams number) composed entirely of the primes in (L119 ) explicitly, it would take about (or at least a not too small fraction of) φ L(1) multiplications mod L(1) (or φ(L) multiplications mod L). Assuming that it takes about one hour for doing 2 · 1010 multiplications mod L(1) or for doing 1010 multiplications mod L on a current PC like mine, it would take about (or at least a not too small fraction of) 10150 =5· 10139 2 · 1010 hours to find a Carmichael number, and about (or at least a not too small fraction of) 10303 =10293 1010 hours to find a Williams number, composed entirely of the primes in Q(L119) explic- itly. There is no known faster algorithm requiring a reasonable amount of memory. However, one might be lucky to construct a Williams number composed of primes NOT entirely in Q(L119) in a reasonable amount of time. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 327 Remark 7.2. Though we could not currently find such a number explicitly, we may foresee something about what such a number looks like. Let · ··· (7.8) n = qi1 qi2 qit be a number with ≥ ··· ≥ 1029 = h 119,4 i1 >i2 > >it 1 (i.e., n is square-free and composed entirely of the primes in Q(L119)). If n is (1) i1 − Carmichael, then 2 1 is greater than (or not much smaller than) φ(L119), thus (1) ≈ the minimum value of i1 is about log2 φ(L ) 497. The number t of divisors of n should satisfy that i i i 1 + 1 + ···+ 1 1 2 t (1) is greater than (or not much smaller than) φ(L119). Then the minimum value of t should happen at i1 = h 119,4 = 1029, thus the minimum value of t should be about 109. This argument comes from the example stated in Remark 7.3. Remark 7.3. Let L = 2345927540754 = 2 · 3 · 7 · 19 · 23 · 79 · 83 · 101 · 193, which is randomly chosen. Then φ(L) = 583560806400 and log2 φ(L)=39.086. It is easy to find that P(L)={p1 n1 = p38 · p32 · p31 · p29 · p27 · p26 · p24 · p23 · p22 · p17 · p15 · p13 · p11 · p9 · p7 · p6 · p5 = 935470 ...043973 (63 digits). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 328 ZHENXIANG ZHANG The number t of divisors of n should satisfy that i i i 1 + 1 + ···+ 1 1 2 t is greater than (or not much smaller than) φ(L). Then the minimum value of t should happen at i1 is equal to (or not much smaller than) |P(L)| = 46, thus the minimum value of t should be greater than (or not much smaller than) 15. Actually, the minimum value of t is equal to 15, which happens at two Carmichael numbers n2 and n3 with i1 is equal to 45 and 46, respectively, coinciding with our estimates: n2 = p45 · p44 · p41 · p32 · p28 · p27 · p24 · p22 · p21 · p20 · p19 · p17 · p16 · p11 · p5 = 326113 ...508372 (63 digits), n3 = p46 · p44 · p41 · p38 · p33 · p31 · p23 · p22 · p20 · p15 · p11 · p8 · p6 · p4 · p2 = 760422 ...374515 (56 digits). This example explains Remark 7.2, though it has no relations with EDONs. Remark 7.4. Analogously, we may foresee something about what a Williams number looks like. Let n be a Williams number composed entirely of the primes in Q(L119), i1 written as in (7.8). Then 2 −1 is greater than (or not much smaller than) φ(L119), ≈ thus the minimum value of i1 is about log2 φ(L119) 1004. The number t of divisors of n should satisfy that i i i 1 + 1 + ···+ 1 1 2 t is greater than (or not much smaller than) φ(L119). Then the minimum value of t should happen when i1 is equal to (or not much smaller than) h 119,4 = 1029, thus the minimum value of t should be about 428. Remark 7.5. Chen and Greene [6] used a variation of the method of Pomerance [15] used to describe sets of primes [6, §3] which, under the equi-distribution hypothesis, one would expect to produce Williams numbers (though they are not called that in the paper). However, their “Williams conductors” are not EDONs, so their mini- mum conjectured “Williams conductor” is much larger than ours. The minimum set of primes they described, which is conjectured to produce Williams numbers, has more than 1600 primes. This remark explains Remark 2.1; see also Remark 5.2. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 329 Supplementary tables Table 8. The first 100 EDONs L with ω = ω(L)andm = |D(L)| No m L ω ei No m L ω ei 1 1 2 1 1 51 576 43243200 6 6321:3 2 2 4 1 2 52 600 64864800 6 5421:3 3 3 8 1 3 53 640 73513440 7 531:5 4 4 12 2 21 54 672 86486400 6 7321:3 5 6 24 2 31 55 720 122522400 7 52:21:4 6 8 48 2 41 56 768 147026880 7 631:5 7 9 72 2 32 57 800 220540320 7 541:5 8 10 96 2 51 58 864 245044800 7 62:21:4 9 12 120 3 31:2 59 896 294053760 7 731:5 10 16 240 3 41:2 60 960 367567200 7 5321:4 11 18 360 3 321 61 1008 490089600 7 72:21:4 12 20 480 3 51:2 62 1024 588107520 7 831:5 13 24 720 3 421 63 1152 735134400 7 6321:4 14 30 1440 3 521 64 1200 1102701600 7 5421:4 15 32 1680 4 41:3 65 1280 1396755360 8 531:6 16 36 2520 4 321:2 66 1344 1470268800 7 7321:4 17 40 3360 4 51:3 67 1440 2205403200 7 6421:4 18 48 5040 4 421:2 68 1536 2793510720 8 631:6 19 60 10080 4 521:2 69 1600 4190266080 8 541:6 20 64 15120 4 431:2 70 1680 4410806400 7 7421:4 21 72 20160 4 621:2 71 1728 4655851200 8 62:21:5 22 80 30240 4 531:2 72 1792 5587021440 8 731:6 23 84 40320 4 721:2 73 1920 6983776800 8 5321:5 24 90 50400 4 52:21 74 2016 9311702400 8 72:21:5 25 96 55440 5 421:3 75 2048 11174042880 8 831:6 26 100 90720 4 541:2 76 2304 13967553600 8 6321:5 27 108 100800 4 62:21 77 2400 20951330400 8 5421:5 28 120 110880 5 521:3 78 2688 27935107200 8 7321:5 29 128 166320 5 431:3 79 2880 41902660800 8 6421:5 30 144 221760 5 621:3 80 3072 55870214400 8 8321:5 31 160 332640 5 531:3 81 3360 83805321600 8 7421:5 32 168 443520 5 721:3 82 3456 97772875200 8 632:21:4 33 180 554400 5 52:21:2 83 3584 128501493120 9 731:7 34 192 665280 5 631:3 84 3600 146659312800 8 542:21:4 35 200 997920 5 541:3 85 3840 160626866400 9 5321:6 36 216 1108800 5 62:21:2 86 4032 195545750400 8 732:21:4 37 224 1330560 5 731:3 87 4096 257002986240 9 831:7 38 240 1441440 6 521:4 88 4320 293318625600 8 642:21:4 39 256 2162160 6 431:4 89 4608 321253732800 9 6321:6 40 288 2882880 6 621:4 90 4800 481880599200 9 5421:6 41 320 4324320 6 531:4 91 5040 586637251200 8 742:21:4 42 336 5765760 6 721:4 92 5376 642507465600 9 7321:6 43 360 7207200 6 52:21:3 93 5760 963761198400 9 6421:6 44 384 8648640 6 631:4 94 6144 1285014931200 9 8321:6 45 400 12972960 6 541:4 95 6720 1927522396800 9 7421:6 46 432 14414400 6 62:21:3 96 6912 2248776129600 9 632:21:5 47 448 17297280 6 731:4 97 7168 3212537328000 9 73:21:6 48 480 21621600 6 5321:3 98 7200 3373164194400 9 542:21:5 49 504 28828800 6 72:21:3 99 7680 3855044793600 9 8421:6 50 512 34594560 6 831:4 100 8064 4497552259200 9 732:21:5 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 330 ZHENXIANG ZHANG Table 9. Some values relative to Ls for 3 ≤ s ≤ 136 s bs ms,1 ln(Ls) αs ms,2 βs εs,0 ei 3 5 4 7.2 0.805993 2 4.9 0.635 521 4 7 6 11.5 0.763090 3 6.7 0.688 62:21 5 11 7 14.1 0.644209 3 7.8 0.561 731:3 6 13 8 18.2 0.621624 3 9.3 0.565 7321:3 7 17 10 22.2 0.593301 4 10.7 0.530 7421:4 8 19 12 27.0 0.586699 4 12.2 0.537 742:21:4 9 23 14 31.6 0.571879 5 13.6 0.517 942:21:5 10 29 16 34.9 0.543903 5 14.6 0.466 942:21:6 11 31 18 39.5 0.533340 5 15.9 0.447 952:21:7 12 37 20 44.3 0.525915 6 17.2 0.435 104321:8 13 41 21 48.0 0.509658 6 18.2 0.402 104321:9 14 43 24 53.0 0.504456 6 19.5 0.393 952:31:9 15 47 25 56.8 0.492022 7 20.5 0.367 952:31:10 16 53 28 63.3 0.496536 7 22.0 0.380 952:41:10 17 59 30 67.4 0.486517 7 23.0 0.358 952:41:11 18 61 32 73.1 0.485278 8 24.5 0.361 9 5 3 2:3 1:12 19 67 34 77.9 0.479211 8 25.5 0.345 9 4:2 2:3 1:13 20 71 36 82.1 0.471667 8 26.5 0.328 9 4:2 2:3 1:14 21 73 39 88.4 0.472217 9 28.0 0.334 11632:31:15 22 79 41 92.7 0.465663 9 29.0 0.319 11632:31:16 23 83 43 97.7 0.461392 10 30.1 0.309 11542:31:17 24 89 46 103.6 0.460339 10 31.4 0.309 1163:22:21:18 25 97 48 108.2 0.455236 10 32.4 0.297 1163:22:21:19 26 101 50 113.7 0.452939 11 33.6 0.292 11632:41:19 27 103 52 119.0 0.450267 11 34.7 0.287 116432:21:21 28 107 54 123.7 0.445989 11 35.7 0.277 116432:21:22 29 109 57 129.3 0.444014 12 36.9 0.273 11642:41:22 30 113 60 136.0 0.444412 12 38.3 0.278 116432:31:23 31 127 62 140.8 0.440811 12 39.3 0.269 116432:31:24 32 131 65 148.6 0.443202 13 40.9 0.279 116432:41:24 33 137 67 153.5 0.439812 13 41.9 0.270 116432:41:25 34 139 69 158.5 0.436590 13 42.9 0.262 116432:41:26 35 149 72 163.5 0.433618 14 43.9 0.255 116432:41:27 36 151 74 169.9 0.433067 14 45.1 0.254 136432:41:28 37 157 77 176.7 0.433067 15 46.5 0.257 116432:51:28 38 163 79 181.8 0.430372 15 47.5 0.250 116432:51:29 39 167 82 186.9 0.427807 15 48.5 0.243 116432:51:30 40 173 84 192.1 0.425379 15 49.5 0.237 116432:51:31 41 179 86 197.2 0.423076 16 50.5 0.232 116432:51:32 42 181 89 204.5 0.423593 16 51.8 0.234 146432:51:33 43 191 92 209.8 0.421426 16 52.8 0.229 146432:51:34 44 193 94 215.0 0.419337 17 53.8 0.224 146432:51:35 45 197 96 220.3 0.417333 17 54.8 0.219 146432:51:36 46 199 99 226.0 0.415865 17 55.9 0.216 137432:51:37 47 211 101 232.3 0.415022 18 57.0 0.214 116432:61:37 48 223 104 237.7 0.413270 18 58.0 0.210 116432:61:38 49 227 106 243.1 0.411580 18 59.0 0.206 116432:61:39 50 229 109 249.2 0.410652 19 60.2 0.204 126432:61:40 51 233 111 254.7 0.409049 19 61.2 0.200 126432:61:41 52 239 113 260.1 0.407509 19 62.2 0.196 126432:61:42 53 241 116 267.0 0.407321 20 63.4 0.197 146432:61:43 54 251 119 272.5 0.405860 20 64.4 0.193 146432:61:44 55 257 122 278.9 0.405178 20 65.5 0.192 128432:61:45 56 263 124 284.5 0.403802 21 66.5 0.189 128432:61:46 57 269 126 290.1 0.402473 21 67.5 0.185 128432:61:47 58 271 129 295.7 0.401176 21 68.5 0.182 128432:61:48 59 277 131 301.3 0.399922 21 69.5 0.179 128432:61:49 60 281 134 307.4 0.399055 22 70.6 0.177 1364:22:61:50 61 283 137 313.8 0.398447 22 71.7 0.176 137532:61:51 62 293 140 321.7 0.398969 23 73.1 0.180 128432:71:51 63 307 143 327.4 0.397825 23 74.1 0.177 128432:71:52 64 311 145 333.2 0.396710 23 75.1 0.174 128432:71:53 65 313 148 338.9 0.395618 23 76.1 0.171 128432:71:54 66 317 151 346.2 0.395618 24 77.4 0.173 1374:22:71:55 67 331 153 352.0 0.394579 24 78.4 0.170 1374:22:71:56 68 337 157 360.6 0.395418 25 79.9 0.175 1284:232:61:57 69 347 160 366.5 0.394406 25 80.9 0.172 1284:232:61:58 70 349 162 372.3 0.393414 25 81.9 0.170 1284:232:61:59 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 331 Table 9 (continued) s bs ms,1 ln(Ls) αs ms,2 βs εs,0 ei 71 353 165 378.2 0.392444 25 82.9 0.167 1284:232:61:60 72 359 167 384.1 0.391499 26 83.9 0.165 1284:232:61:61 73 367 170 391.0 0.391167 26 85.0 0.165 1574:232:61:62 74 373 173 396.9 0.390261 26 86.0 0.162 1574:232:61:63 75 379 175 402.8 0.389378 27 87.0 0.160 1574:232:61:64 76 383 178 409.4 0.388868 27 88.1 0.160 1285432:61:65 77 389 181 415.4 0.388017 27 89.1 0.158 1285432:61:66 78 397 184 421.8 0.387445 28 90.2 0.156 1584:232:61:67 79 401 187 429.3 0.387446 28 91.4 0.158 1284:232:71:67 80 409 190 437.0 0.387479 28 92.7 0.159 1285432:71:68 81 419 193 443.0 0.386679 29 93.7 0.157 1285432:71:69 82 421 196 449.0 0.385892 29 94.7 0.155 1285432:71:70 83 431 198 455.1 0.385127 29 95.7 0.153 1285432:71:71 84 433 201 461.2 0.384373 30 96.7 0.151 1285432:71:72 85 439 203 467.3 0.383636 30 97.7 0.150 1285432:71:73 86 443 207 474.7 0.383567 30 98.9 0.150 1485432:71:74 87 449 209 480.9 0.382847 31 99.9 0.149 1485432:71:75 88 457 212 487.0 0.382144 31 100.9 0.147 1485432:71:76 89 461 216 495.4 0.382502 31 102.3 0.149 1285432:81:76 90 463 218 501.6 0.381805 32 103.3 0.148 1285432:81:77 91 467 221 507.7 0.381120 32 104.3 0.146 1285432:81:78 92 479 224 514.1 0.380556 32 105.3 0.145 148543:22:61:80 93 487 226 520.3 0.379902 33 106.3 0.144 148543:22:61:81 94 491 230 527.7 0.379739 33 107.5 0.144 1485432:81:81 95 499 232 533.9 0.379103 33 108.5 0.142 1485432:81:82 96 503 236 542.5 0.379438 34 109.9 0.145 1285432:91:82 97 509 239 548.7 0.378810 34 110.9 0.143 1285432:91:83 98 521 242 555.0 0.378198 34 111.9 0.142 1285432:91:84 99 523 245 562.1 0.377936 35 113.0 0.142 158543:22:71:86 100 541 248 568.9 0.377538 35 114.1 0.141 1485432:91:86 101 547 251 575.9 0.377217 35 115.2 0.141 1585432:91:87 102 557 254 584.8 0.377594 36 116.6 0.143 158543:22:81:88 103 563 257 591.1 0.377018 36 117.6 0.142 158543:22:81:89 104 569 260 597.5 0.376451 36 118.6 0.140 158543:22:81:90 105 571 263 603.8 0.375892 37 119.6 0.139 158543:22:81:91 106 577 266 610.2 0.375341 37 120.6 0.138 158543:22:81:92 107 587 268 616.5 0.374802 37 121.6 0.137 158543:22:81:93 108 593 271 622.9 0.374270 37 122.6 0.135 158543:22:81:94 109 599 274 629.3 0.373748 38 123.6 0.134 158543:22:81:95 110 601 277 635.7 0.373230 38 124.6 0.133 158543:22:81:96 111 607 281 646.0 0.373991 39 126.2 0.137 158543:22:91:96 112 613 284 652.4 0.373474 39 127.2 0.136 158543:22:91:97 113 617 287 658.8 0.372964 39 128.2 0.134 158543:22:91:98 114 619 289 665.3 0.372460 39 129.2 0.133 158543:22:91:99 115 631 292 671.7 0.371966 40 130.2 0.132 158543:22:91:100 116 641 295 678.2 0.371482 40 131.2 0.131 158543:22:91:101 117 643 298 684.6 0.371002 40 132.2 0.130 158543:22:91:102 118 647 301 691.1 0.370529 41 133.2 0.129 158543:22:91:103 119 653 304 698.8 0.370437 41 134.4 0.129 14 8 5:2 3:2 2:9 1:104 120 659 307 705.3 0.369973 41 135.4 0.128 14 8 5:2 3:2 2:9 1:105 121 661 310 711.8 0.369513 42 136.4 0.127 14 8 5:2 3:2 2:9 1:106 122 673 312 718.3 0.369062 42 137.4 0.126 14 8 5:2 3:2 2:9 1:107 123 677 315 724.9 0.368616 42 138.4 0.125 14 8 5:2 3:2 2:9 1:108 124 683 319 734.1 0.368946 43 139.8 0.127 158543:22:101:108 125 691 322 740.6 0.368505 43 140.8 0.126 158543:22:101:109 126 701 325 747.2 0.368072 43 141.8 0.125 158543:22:101:110 127 709 328 753.8 0.367645 43 142.8 0.124 158543:22:101:111 128 719 331 760.3 0.367224 44 143.8 0.123 158543:22:101:112 129 727 334 766.9 0.366810 44 144.8 0.122 158543:22:101:113 130 733 336 773.5 0.366402 44 145.8 0.121 158543:22:101:114 131 739 340 782.1 0.366509 45 147.0 0.122 15 8 5:2 3:2 2:10 1:115 132 743 343 788.7 0.366104 45 148.0 0.121 15 8 5:2 3:2 2:10 1:116 133 751 346 795.3 0.365706 45 149.0 0.121 15 8 5:2 3:2 2:10 1:117 134 757 349 801.9 0.365312 46 150.0 0.120 15 8 5:2 3:2 2:10 1:118 135 761 352 808.6 0.364922 46 151.0 0.119 15 8 5:2 3:2 2:10 1:119 136 769 355 815.2 0.364537 46 152.0 0.118 15 8 5:2 3:2 2:10 1:120 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 332 ZHENXIANG ZHANG Table 10. The set Q(L119) i qi i qi i qi i qi i qi 1 787 61 92179 121 972823 181 5825899 241 19409239 2 907 62 93199 122 977023 182 6671323 242 20414899 3 1087 63 96223 123 1013239 183 6940663 243 21919999 4 1279 64 98779 124 1078183 184 7052599 244 22974199 5 1459 65 100279 125 1135699 185 7092667 245 23156767 6 1759 66 101323 126 1146727 186 7273099 246 23689999 7 1867 67 109663 127 1386043 187 7342903 247 23942143 8 1999 68 115183 128 1409227 188 7349239 248 24101239 9 2383 69 130807 129 1411099 189 7354639 249 24560299 10 2503 70 144439 130 1472719 190 7428079 250 25514239 11 2647 71 154087 131 1474999 191 7648327 251 25886407 12 2719 72 158047 132 1498303 192 7879999 252 27878527 13 3319 73 160687 133 1502047 193 7924879 253 28654663 14 3607 74 170179 134 1548739 194 8016139 254 29097199 15 3739 75 203659 135 1555999 195 8159659 255 29368159 16 4219 76 208279 136 1657099 196 8162743 256 29786719 17 5059 77 226027 137 1681027 197 8222959 257 31135939 18 5779 78 232207 138 1852579 198 8570719 258 33603379 19 5839 79 241543 139 1900687 199 8633899 259 34543699 20 6163 80 245527 140 2104867 200 8756203 260 35295583 21 6679 81 247339 141 2111839 201 8942383 261 35504383 22 7687 82 262399 142 2129599 202 9139807 262 36177199 23 7879 83 268927 143 2151847 203 9247123 263 36201439 24 9007 84 271363 144 2327599 204 9297367 264 36692839 25 9343 85 273943 145 2404819 205 9378559 265 38763019 26 9739 86 284407 146 2539567 206 9594799 266 39760687 27 12799 87 295039 147 2744323 207 9613567 267 40613407 28 13183 88 296479 148 2859559 208 9700147 268 40927303 29 15583 89 318007 149 2878039 209 10023679 269 41835067 30 16759 90 327343 150 2964319 210 10026019 270 41977279 31 18127 91 346543 151 2988259 211 10034623 271 42808039 32 20359 92 355027 152 2988343 212 10246783 272 43070779 33 21799 93 395119 153 3055963 213 10346239 273 46471699 34 25747 94 402379 154 3190519 214 10348399 274 46887499 35 30727 95 408703 155 3349999 215 10481563 275 47510443 36 32119 96 426799 156 3355423 216 10485427 276 47756287 37 34543 97 434659 157 3511999 217 10706599 277 47913343 38 36187 98 479599 158 3576583 218 11006647 278 48389239 39 38959 99 496399 159 3637279 219 11112319 279 48826243 40 40639 100 519247 160 3671599 220 11480743 280 48957703 41 42283 101 536923 161 3728719 221 11615419 281 48977947 42 44179 102 565723 162 3739999 222 12453043 282 49097179 43 45259 103 581743 163 3741823 223 12587563 283 49876219 44 47287 104 622423 164 3819943 224 13049479 284 50723527 45 49999 105 627919 165 3920743 225 13306303 285 54220363 46 50383 106 629263 166 3926959 226 13608379 286 55149823 47 50839 107 645739 167 3999283 227 13914559 287 56566567 48 54559 108 669943 168 4019047 228 14431999 288 57102319 49 55807 109 689599 169 4170583 229 14519359 289 57325087 50 63499 110 731767 170 4631359 230 14683267 290 57667807 51 65599 111 756919 171 4697599 231 14710099 291 57975499 52 68059 112 767359 172 4708999 232 16048699 292 62155999 53 70183 113 771643 173 4742047 233 16062463 293 64069279 54 73699 114 772339 174 5078347 234 16220263 294 64803727 55 74719 115 781999 175 5477599 235 16980559 295 65823223 56 77263 116 853903 176 5551999 236 17210467 296 66681907 57 77659 117 872959 177 5593183 237 17613823 297 67595599 58 78823 118 912367 178 5685499 238 17697019 298 69826543 59 80239 119 934579 179 5693899 239 18183799 299 70407199 60 90403 120 961927 180 5803003 240 19279999 300 70429339 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 333 Table 10 (continued) i qi i qi i qi i qi i qi 301 70618447 361 241647199 421 861576319 481 2228322799 541 5982065623 302 74593699 362 244584667 422 864315847 482 2245353559 542 6030530767 303 74993959 363 257961559 423 868374979 483 2246597179 543 6111249259 304 75673999 364 258083803 424 882386299 484 2249132719 544 6220227679 305 76630387 365 267596863 425 894146527 485 2286649279 545 6274334719 306 81601087 366 305205247 426 912630079 486 2327261899 546 6280423423 307 82062319 367 311009599 427 933993667 487 2363974783 547 6541791079 308 83649499 368 311308447 428 954712747 488 2389614319 548 6592100299 309 84144763 369 324688099 429 959153599 489 2412969283 549 6685541059 310 84572503 370 325743799 430 973312447 490 2482211443 550 6858726439 311 89514559 371 329547583 431 988410463 491 2502142099 551 6913369807 312 90510967 372 333142759 432 996707407 492 2506949719 552 7003264927 313 95571199 373 335205127 433 1018347079 493 2542573087 553 7386691327 314 97399999 374 335288479 434 1026768247 494 2558642899 554 7582915999 315 98866279 375 339283039 435 1033908643 495 2603261467 555 7650427999 316 102007459 376 351873187 436 1034352127 496 2701857799 556 7740919999 317 103288399 377 368475199 437 1040140159 497 2755425499 557 7930031383 318 105277807 378 375079363 438 1046722639 498 2872740859 558 7982388919 319 106217407 379 378513523 439 1062512263 499 2893373683 559 8016535339 320 106831999 380 402949639 440 1064287927 500 3124754119 560 8328861139 321 111051103 381 417392419 441 1102611967 501 3194259199 561 8704635187 322 111336499 382 422843167 442 1134835759 502 3197022223 562 8956524799 323 112445503 383 452403199 443 1173887527 503 3308376283 563 8988956167 324 113288059 384 455233003 444 1180092187 504 3353904079 564 9143925499 325 120071767 385 461335423 445 1216474099 505 3375797647 565 9828203743 326 125534683 386 469890559 446 1253152699 506 3484274683 566 10016115127 327 126018439 387 477136747 447 1267785919 507 3549697399 567 10254236959 328 126339487 388 487393939 448 1271322739 508 3556130527 568 10271167999 329 127100023 389 500913379 449 1276181407 509 3676318627 569 10325467279 330 132640639 390 505029799 450 1280126179 510 3698782207 570 10900965679 331 135491539 391 525392479 451 1303701439 511 3711108607 571 11003827999 332 137569759 392 547081039 452 1339453207 512 3723161599 572 11016717007 333 138312679 393 551098123 453 1352806879 513 3754108639 573 11113502479 334 144082303 394 553689319 454 1356079999 514 3779811199 574 11298517423 335 144087679 395 554021599 455 1417793599 515 3790388683 575 11449429759 336 147243799 396 560810407 456 1422866719 516 3823169239 576 11512651879 337 148713199 397 568404799 457 1445946763 517 3898188799 577 11574409879 338 148915279 398 576053623 458 1454402359 518 3980812543 578 11913769999 339 154506367 399 582166063 459 1473221023 519 4051400767 579 12605227519 340 155439103 400 595354999 460 1485028819 520 4051760227 580 12846819259 341 158176999 401 602670007 461 1558078579 521 4190060599 581 12846848563 342 163352863 402 610215667 462 1617131839 522 4256785807 582 13109614939 343 167656519 403 619166719 463 1620340039 523 4278998599 583 13272405199 344 169642999 404 629423299 464 1634234743 524 4359232099 584 13428532303 345 180840703 405 629491999 465 1683539443 525 4483248043 585 13614455359 346 184483807 406 649265839 466 1717278559 526 4507477759 586 13813869247 347 184611799 407 659668699 467 1720464943 527 4545949999 587 13947454459 348 185798887 408 690641239 468 1833210367 528 4591373599 588 14065003423 349 187944499 409 714583039 469 1839914623 529 4595759299 589 14232579103 350 192747943 410 757529599 470 1858679239 530 4795965679 590 14397373999 351 196349119 411 764211199 471 1865544367 531 4846277479 591 14614349023 352 197503639 412 773068063 472 1922259907 532 4887418303 592 14971910539 353 199180459 413 795335503 473 1924656007 533 4947190399 593 15339536803 354 202295059 414 804725503 474 1960737847 534 4980700063 594 15392580799 355 203919103 415 810674959 475 2002150879 535 5153031679 595 15461200759 356 207601039 416 815820799 476 2035082239 536 5286546703 596 15472540903 357 211870999 417 824286403 477 2046777727 537 5330208703 597 15660894439 358 213161167 418 828270847 478 2080785799 538 5745497119 598 15851289079 359 222361759 419 846632299 479 2090595907 539 5754151999 599 16067631259 360 230829103 420 850685599 480 2159566939 540 5925039919 600 17165035519 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 334 ZHENXIANG ZHANG Table 10 (continued) i qi i qi i qi i qi 601 17356240567 661 54935089483 721 157598233759 781 639555769399 602 17670844927 662 55090574047 722 158408319343 782 664525725319 603 18067933759 663 55437587479 723 159489712639 783 673734369439 604 18252049999 664 55512893983 724 168930606079 784 678277343479 605 18380038423 665 56247131599 725 173209759243 785 686982319999 606 18574730899 666 57275420119 726 180253602187 786 716485407499 607 18581930347 667 57468126319 727 181163828119 787 744646072447 608 18656323327 668 58032761599 728 181519820479 788 832267812379 609 18866154799 669 58114728127 729 186456219223 789 858816720199 610 18911487799 670 58476301003 730 187485511039 790 884208774079 611 19126773163 671 58603195999 731 192927891907 791 888642199999 612 19131428863 672 59104286719 732 195873987583 792 902455259599 613 19203269023 673 60912408523 733 196263830443 793 909135539443 614 19444050487 674 66231779167 734 201981439999 794 966535032439 615 19608303079 675 66478615327 735 209151316879 795 1023043486207 616 19752622159 676 67076977879 736 212362130023 796 1023161149999 617 20150730199 677 69625729447 737 213119398399 797 1056513909439 618 20320030207 678 71339913499 738 224118139519 798 1071014325967 619 20415115099 679 71346148147 739 227636235919 799 1099500892687 620 20853225079 680 73426847167 740 232117234519 800 1119151553599 621 22346370463 681 76762686079 741 245156892163 801 1149328945927 622 22676077999 682 78172876927 742 250966894159 802 1168037428699 623 23544646399 683 79760151199 743 251226024499 803 1190409468319 624 24046564123 684 80578592767 744 262293817087 804 1223875523599 625 24504589999 685 80721643999 745 279867875407 805 1232329254859 626 26119521199 686 81012649999 746 280440602899 806 1232905982527 627 26517413119 687 81750512299 747 286248077119 807 1287674115127 628 26662983439 688 83123389183 748 288656629759 808 1392889422907 629 27394815703 689 83449833199 749 293624864107 809 1431188836543 630 29624790919 690 83805628483 750 314255912959 810 1642103213899 631 29651398519 691 86665577239 751 322316148799 811 1698234710203 632 29789361487 692 86814086347 752 331447374847 812 1703060330479 633 30261425839 693 89997112999 753 339849995479 813 1719613786879 634 30691198903 694 95598739939 754 353256933199 814 1774266600247 635 31484725759 695 103262638159 755 373472861959 815 1869760249807 636 31778872399 696 104589411007 756 389829736999 816 1873597610179 637 32360924779 697 105266990287 757 394246803499 817 1952913722803 638 34356679999 698 105427154359 758 394766260579 818 1988338913263 639 34673435959 699 108135017407 759 411210498379 819 2025033640519 640 35459747839 700 109310496283 760 412468198399 820 2037941876143 641 37306426327 701 109956041479 761 413379968779 821 2062268708959 642 37498501603 702 110999417539 762 416754219199 822 2136163904239 643 37542237643 703 111027558559 763 425039717467 823 2142958069999 644 39268720639 704 116429504959 764 437914378399 824 2211718788739 645 40367255839 705 118626572899 765 445943869099 825 2255852561383 646 40841131423 706 121880363299 766 446208291223 826 2264762086399 647 41201999359 707 122721308599 767 457241283019 827 2286189631027 648 41526861559 708 125688862639 768 459458195239 828 2308568099023 649 42314953339 709 127966423807 769 491702851039 829 2349898797619 650 42804567919 710 129691452367 770 498619994239 830 2354896162207 651 43794278263 711 143317497727 771 537415171039 831 2364829939327 652 44703953743 712 144542248399 772 541390950907 832 2408159379007 653 46793887687 713 145773715699 773 545252536519 833 2517617340967 654 47663492563 714 145941637339 774 549386287999 834 2578569831199 655 48112650379 715 150214489699 775 551733590539 835 2580566833099 656 49573283839 716 153572531743 776 568258266619 836 2715678042127 657 50582652079 717 153858759679 777 593169271423 837 2731686889663 658 50804766559 718 154333913599 778 598955884927 838 2760706582999 659 52670343043 719 155922027007 779 631329991363 839 2888449327999 660 53322350047 720 156392587999 780 636489301519 840 2929942151839 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ESTIMATING THE COUNTS OF CARMICHAEL AND WILLIAMS NUMBERS 335 Table 10 (continued) i qi i qi i qi 841 3033753331519 901 26123931426367 961 752446947492199 842 3110539894723 902 29960137749019 962 820128813447679 843 3227460532999 903 29987026558399 963 827402716617439 844 3474156616639 904 32861084601223 964 842550848470399 845 3596892544519 905 34299794693119 965 920134184163307 846 4051667592463 906 35460827837839 966 923730153783139 847 4106550714319 907 36837608241799 967 995579871938263 848 4161489723379 908 37579466912599 968 1009585315548499 849 4165541963923 909 38881942887259 969 1058682488556499 850 4308360366259 910 39063694303999 970 1070330498689999 851 4460878906303 911 41819983412719 971 1144487909473039 852 4528550605567 912 42590019487039 972 1148243238308239 853 4594036387999 913 42844598213119 973 1280439916463359 854 4672436218507 914 44017837846399 974 1413599893343527 855 4909398940387 915 44716356862399 975 1446745424007487 856 5031304789759 916 48901053543967 976 1640361138311143 857 5214218580607 917 49439182551679 977 1809681632680639 858 5226824061967 918 49558806033283 978 1879862198962879 859 5695072326499 919 52059347400499 979 1883040830658919 860 5893429527199 920 52783968299119 980 1935212300541343 861 6141440416879 921 54166906892647 981 2170815986212147 862 6248394822259 922 56136592317079 982 2857501249327999 863 6302317191847 923 61223107804699 983 3012439275199999 864 6643697457307 924 87522006260347 984 3588349381405939 865 7491784717999 925 94699490227759 985 4140976705075099 866 7716028564063 926 95730899453743 986 4869186874946899 867 7805643198679 927 96328285383919 987 5411613836417023 868 7936587955999 928 97653606499699 988 6073624238445499 869 8830349269939 929 103472325171619 989 6970408885351999 870 9060408318799 930 108533627170627 990 6987654632824399 871 9106360231099 931 109393975102987 991 7132074358352959 872 9680211926659 932 111549780234499 992 7180712804095423 873 9971196577759 933 120716853828559 993 7832107985091199 874 10793617362439 934 141754843999039 994 9385146881412499 875 10795626776887 935 161057759781907 995 9830337648422587 876 11684401579999 936 167987799007219 996 10712979037583299 877 12426091519999 937 169027288307767 997 12948212716519999 878 13447939846087 938 170821211109823 998 15192453921925939 879 14020313382199 939 173574428913439 999 21505211481904543 880 14297950849159 940 180157987835407 1000 31532582453744503 881 14769480803899 941 216256971057499 1001 35579917868667439 882 15664721014099 942 246251700244747 1002 40646000733518839 883 16007673073663 943 251646857809327 1003 42826288958497279 884 16104305851999 944 268141900279807 1004 45823901453072263 885 16153432198543 945 301465674559999 1005 45876495399838399 886 16902327305119 946 309141173811499 1006 58247207359323439 887 17186810663647 947 319147129378303 1007 75721286224664263 888 17500211323759 948 319396069833283 1008 80509305814198759 889 18012753709939 949 361621903554559 1009 82076095638158599 890 18811100389999 950 430161439634599 1010 95767121062884259 891 20176437318679 951 569172865847167 1011 112054809497633419 892 20366774971999 952 582291720686719 1012 115541716197093499 893 20604299477959 953 597342766039027 1013 128059275462081499 894 22210034852323 954 626698249018819 1014 155063736548762623 895 23937810469759 955 693540047264899 1015 220964467807900639 896 24060759055999 956 699632470659487 1016 229731074792916607 897 24062544244639 957 701376494677279 1017 238429710488061667 898 24375516432079 958 716113052098399 1018 306781581482781679 899 24405610912159 959 738872523704383 1019 357598761931630207 900 24505851952999 960 746468318157283 1020 369698388689722639 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 336 ZHENXIANG ZHANG Table 10 (continued) i qi qi − 1 1021 476535548296332799 2 · 3 · 7 · 13 · 53 · 89 · 107 · 113 · 163 · 223 · 421 1022 604316767944367999 2 · 34 · 132 · 37 · 71 · 199 · 337 · 349 · 359 1023 852632436484598527 2 · 32 · 71 · 113 · 131 · 383 · 443 · 491 · 541 1024 6621174447229245919 2 · 3 · 7 · 19 · 29 · 101 · 107 · 271 · 359 · 503 · 541 1025 26379015464291154499 2 · 32 · 19 · 61 · 151 · 281 · 293 · 359 · 491 · 577 1026 33843204175351651039 2 · 33 · 7 · 19 · 53 · 101 · 131 · 139 · 311 · 359 · 433 1027 224489273360340970879 2 · 3 · 13 · 19 · 101 · 107 · 199 · 397 · 443 · 619 · 647 1028 4194019937445612397579 2 · 38 · 72 · 13 · 19 · 397 · 433 · 443 · 541 · 641 1029 4905729483343410746599 2 · 3 · 72 · 192 · 37 · 101 · 311 · 337 · 433 · 479 · 569 i qi +1 1021 29 · 52 · 11 · 97 · 241 · 461 · 509 · 617 1022 27 · 53 · 31 · 137 · 167 · 257 · 353 · 587 1023 28 · 11 · 172 · 41 · 167 · 467 · 547 · 599 1024 25 · 5 · 47 · 109 · 233 · 331 · 431 · 487 · 499 1025 22 · 53 · 11 · 31 · 67 · 73 · 313 · 379 · 467 · 571 1026 25 · 5 · 137 · 149 · 241 · 277 · 467 · 509 · 653 1027 27 · 5 · 41 · 47 · 97 · 103 · 233 · 307 · 487 · 523 1028 22 · 5 · 67 · 73 · 97 · 179 · 191 · 257 · 331 · 379 · 401 1029 23 · 52 · 312 · 67 · 73 · 137 · 257 · 353 · 643 · 653 Acknowledgment The author thanks the referees for kind and helpful comments that improved the presentation of this paper. 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We have by Lemmas 3.2 and 3.4, ⇒ − 1 − ej =2= uj = e1/4 >e1/4 1 > log2 bs 1 4 bs+1 2bs+1 2bs+1 =⇒ bj < < =⇒ j = π(bj ) ≤ π . uj 1/4 1/4 2 bs bs Since [8] x 1.2762 π(x) ≤ 1+ for x > 1 ln x ln x and (3.6) and (3.7), a simple computation shows that 2b π s+1