MATHEMATICS OF COMPUTATION Volume 76, Number 260, October 2007, Pages 2095–2107 S 0025-5718(07)01977-1 Article electronically published on April 17, 2007
TWO KINDS OF STRONG PSEUDOPRIMES UP TO 1036
ZHENXIANG ZHANG
Dedicated to the memory of Kencheng Zeng (1927–2004)
Abstract. Let n>1 be an odd composite integer. Write n − 1=2sd with d r odd. If either bd ≡ 1modn or b2 d ≡−1modn for some r =0, 1,...,s− 1, then we say that n isastrongpseudoprimetobaseb, or spsp(b)forshort. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If we know the exact value of ψt, we will have, for integers n<ψt, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψt are known for 1 ≤ t ≤ 8. Conjectured values of ψ9,...,ψ12 were given by us in our previous papers (Math. Comp. 72 (2003), 2085–2097; 74 (2005), 1009–1024). ≤ ≤ The main purpose of this paper is to give exact values of ψt for 13 t 19; 36 to give a lower bound of ψ20: ψ20 > 10 ; and to give reasons and numerical 36 evidence of K2- and C3-spsp’s < 10 to support the following conjecture: ≥ ψt = ψt <ψt for any t 12, where ψt (resp. ψt ) is the smallest K2- (resp. C3-) strong pseudoprime to all the first t prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp’s < 1036 to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: n = pq with p, q primes and q − 1=2(p − 1); and that a C3-spsp is an spsp and a Carmichael number of the form: n = q1q2q3 with each prime factor qi ≡ 3mod4.)
1. Introduction Let n>1 be an odd integer. Write n − 1=2sd with d odd. We say that n passes the Miller (strong probable prime) test [5] to base b,orthatn is an sprp(b) for short, if r (1.1) either bd ≡ 1(modn)orb2 d ≡−1(modn)forsomer =0, 1,...,s− 1. (The original test of Miller [5] was somewhat more complicated and was a deter- ministic, ERH-based test; see [1, Section 3.4].) If n is composite and (1.1) holds, then we say that n is a strong pseudoprime to base b, or spsp(b) for short. An spsp(b1,...,bt)isanspsptoallthet bases. Define SB(n) (1.2) SB(n)=#{b ∈ Z :1≤ b ≤ n − 1,n is an spsp(b)} and P (n)= , R ϕ(n)
Received by the editor March 8, 2006 and, in revised form, July 8, 2006. 2000 Mathematics Subject Classification. Primary 11Y11, 11A15, 11A51. Key words and phrases. K2-strong pseudoprimes, C3-strong pseudoprimes, the least strong pseudoprime to the first t prime bases, Rabin-Miller test, biquadratic residue characters, Chinese remainder theorem. The author was supported by the NSF of China Grant 10071001.
c 2007 American Mathematical Society Reverts to public domain 28 years from publication 2095
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where ϕ is Euler’s function. Monier [6] and Rabin [8] proved that if n is an odd com- posite positive integer, then SB(n) ≤ (n−1)/4. In fact, as pointed out by Damg˚ard, Landrock and Pomerance [2], if n = 9 is odd and composite, then SB(n) ≤ ϕ(n)/4, i.e., PR(n) ≤ 1/4. These facts lead to the Rabin-Miller test: given a positive integer n,pickk different positive integers less than n and perform the Miller test on n for each of these bases; if n is composite, the probability that n passes all k tests is less than 1/4k. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If n<ψt, then only t Miller tests are needed to find out whether n is prime or not. This means that if we know the exact value of ψt, then for integers n<ψt we will have a deterministic primality testing algorithm which is not only easier to implement but also faster than existing deterministic primality testing algorithms. From Pomerance et al. [7] and Jaeschke [4] we know the exact values of ψt for 1 ≤ t ≤ 8 and upper bounds for ψ9, ψ10 and ψ11:
ψ9 ≤ 41234 31613 57056 89041 (20 digits) = 4540612081 · 9081224161,
ψ10 ≤ 155 33605 66073 14320 55410 02401 (28 digits) = 22754930352733 · 68264791058197,
ψ11 ≤ 5689 71935 26942 02437 03269 72321 (29 digits) = 137716125329053 · 413148375987157. Jaeschke [4] tabulated all strong pseudoprimes < 1012 to the bases 2, 3, and 5. There are in total 101 of them. Among these 101 numbers there are 95 numbers n having the form (1.3) n = pq with p, q odd primes and q − 1=k(p − 1), with k ∈{2, 3, 4, 5, 6, 7, 13, 4/3, 5/2}; the other six numbers are Carmichael numbers with three prime factors in the sense that:
(1.4) n = q1q2q3 with q1 ψ11 ≤ N11 = 73 95010 24079 41207 09381 (22 digits) = 60807114061 · 121614228121, and a 24-digit upper bound for ψ12 was obtained: ψ12 ≤ N12 = 3186 65857 83403 11511 67461 (24 digits) (1.5) = 399165290221 · 798330580441. In [11], we first followed our previous work [9] to find all K4/3-, K5/2-, K3/2-, K6-spsp’s < 1024 to the first several prime bases. No spsp’s of such forms to the first 8 prime bases are found. Note that the three bounds N10,N11 and N12 are all License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRONG PSEUDOPRIMES TO 1036 2097 K2-spsp’s with PR(n)=3/16. These facts give us a hint that to lower these upper bounds, we should find those numbers n with PR(n) equal to or close to 1/4. For short, we call a Carmichael number n = q1q2q3 with each prime factor qi ≡ 3 mod 4 a C3-number. If n is a C3-number and an spsp(b1,b2,...,bt), we call n a C3-spsp(b1,b2,...,bt). It is easy to prove that (see [11, §5]) PR(n)=1/4 ⇐⇒ (1.6) either n = pq is a K2-number with p ≡ 3mod4 or n is a C3-number; and that (1.7) if n is an spsp(2), then PR(n)=1/4 ⇐⇒ n is a C3-number. We [11] then focused our attention to develop a method for finding all C3-spsp(2, 3, 20 5, 7, 11)’s < 10 . As a result the upper bounds for ψ9,ψ10 and ψ11 are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number: ψ9 ≤ ψ10 ≤ ψ11 ≤ Q11 = 3825 12305 65464 13051 (19 digits) = 149491 · 747451 · 34233211. We [11] at last gave reasons to support the following Conjecture 1 (see also [3, Problem A12]). Conjecture 1. We have ψ9 = ψ10 = ψ11 = 3825 12305 65464 13051 (19 digits). Let q1 Then we call d the kernel, the triple (h1,h2,h3)thesignature,andH = h1h2h3 the height of N, respectively. In [10, Section 2], we described a procedure for finding C3-spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 C3- spsp(2)’s < 1024 with heights < 109, only three of which are spsp’s to the first 11 24 prime bases up to 31. No C3-spsp’s < 10 to the first 12 prime bases with heights < 109 were found. Denote by ψt (resp. ψt ) the smallest K2- (resp. C3-) spsp to all the first t prime bases. In [10, §5], we gave reasons to support the following Conjecture 2. Conjecture 2. We have 24 (1.8) ψ12 = ψ12 < 10 <ψ12, where ψ12 = N12 = 3186 65857 83403 11511 67461 (24 digits) = 399165290221 · 798330580441 was found in [9] (see (1.5)). The main purpose of this paper is to give reasons and numerical evidence of K2- 36 and C3- strong pseudoprimes < 10 to support the following Conjecture 3, where ≤ ≤ the exact values of ψt for 13 t 19 and an upper bound of ψ20 are given in the following Proposition 1.1. Conjecture 3. We have ψt = ψt <ψt for any t ≥ 12. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2098 ZHENXIANG ZHANG Remark 1.1. Conjecture 3 covers Conjecture 2, which is the case t = 12. Proposition 1.1. We have ψ13 = 33170 44064 67988 73859 61981 (25 digits) = 1287836182261 · 2575672364521; ψ14 = 600 30942 89670 10580 03125 96501 (28 digits) = 54786377365501 · 109572754731001; ψ15 = 5927 63610 75595 57326 34463 30101 (29 digits) = 172157429516701 · 344314859033401; ψ16 = ψ17 = 56413 29280 21909 22101 40875 01701 (30 digits) = 531099297693901 · 1062198595387801; ψ18 = ψ19 = 1543 26786 44434 20616 87767 76407 51301 (34 digits) = 27778299663977101 · 55556599327954201; 36 ψ20 > 10 . In Section 2 we describe a procedure for finding all K2-spsp’s 2. K2-strong pseudoprimes up to 1036 Let π be a primary irreducible of the ring Z[i] of Gaussian integers such that q = ππ ≡ 1mod4andp =(q +1)/2 are two primes determined by π.Denoteby b π 4 the biquadratic residue character symbol of b modulo π.Putpα =(αα +1)/2 for α ∈ Z[i]. Let 2 − xy pα 1 R2 = primary α = x + yi :0≤ x, y < 8,i 2 =(−1) 8 = {1, 5+4i}. Foraprimeb ≡ 3mod4,let α pα Rb = α = x + yi :0≤ x, y < 4b, α ≡ 1mod4, = ; b 4 b and for a prime b ≡ 1mod4,let αα−1 p R = α = x + yi :0≤ x, y < 4b, α ≡ 1mod4, = α , b β b 4 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRONG PSEUDOPRIMES TO 1036 2099 ∗ (t) t where ∗ is the Jacobi symbol. Let M =4 j=1 bj,wherebj is the jth prime. Applying the Chinese Remainder Theorem, it is easy to compute the set R(t) = x + yi :0≤ x, y < M (t), x + yi (mod 4b) ∈ Rb for all the first t prime bases b . In [9], we described a procedure to compute all K2-numbers n = pq, below a given limit L (say 1024), which are strong pseudoprimes to the first t ≥ 6 prime bases. The procedure is based on the following proposition. Proposition 2.1 ([9, Proposition 3.2]). If n = pq is an spsp to the first t prime bases, then there exists α ∈ R(t) such that π ≡ α mod M (t). Since M (6) =4·2·3·5·7·11·13 = 120120 and #R(6) =2·2·2·12·30·30 = 86400 are of suitable size for programming on a PC486 with Turbo pascal 6.0, we [9] successfully found all K2-strong pseudoprimes < 1024 to the first six prime bases. Now our objective is to compute all K2-numbers n = pq <1036 on a PC Pentium IV/1.8 GHz with Delphi 6.0, which are strong pseudoprimes to the first 9 prime bases. To speed things up, we should use a larger database R(t). However, since #R(9) =#R(6) · 56 · 90 · 132 = 4838400 · 11880 = 57480192000, the set R(9) is too large to fit in either memory or in a disk file. Considering the storage requirements and the efficiency of the algorithm, we pre-compute the set R(7) and the set S = x + yi :0≤ x, y < 4 · 19 · 23,x+ yi (mod 4b) ∈ Rb for b =19and23 with M (7) = M (6) · 17 = 2042040, #R(7) =#R(6) · 56 = 4838400 and #S =90· 132 = 11880. Now we are ready to describe a procedure to compute all K2-spsp’s PROCEDURE 2.1. Finding K2-spsp to the first t (≥ 9) prime bases; BEGIN (7) For every x1 + y1i ∈ R Do For every x2 + y2i ∈ S Do begin Using the CRT, compute x and y such that 0 ≤ x, y < M (9),x+ yi (mod M (7)) ∈ R(7) and x + yi (mod 4 · 19 · 23) ∈ S; √ ≥ ≥ ≤ 4 8L For u 0,v 0,u+ v M (9) +1Do Begin 2 2 q ← x + uM (9) + y + vM(9) ; p ← (q +1)/2; n ← p · q; If n is an spsp to the first t prime bases Then output n, p and q; 2 2 q ← x − uM (9) + y + vM(9) ; p ← (q +1)/2; n ← p · q; If n is an spsp to the first t prime bases Then output n, p and q End end END. The Delphi-Pascal program (with multi-precision package partially written in Assembly language) ran about 2400 hours on a PC Pentium IV/1.8GHz (in fact we License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2100 ZHENXIANG ZHANG used 10 PCs with each running 10 days) to get all K2-spsp’s < 1036 to the first nine prime bases 2, 3, 5, 7, 11, 13, 17, 19, and 23. There are in total 90002828 numbers, 100920 of which are spsp’s to the first 13 prime bases, 24 numbers are spsp’s to the first 18 prime bases up to 61 (listed in Table 1), 4 numbers are spsp’s to the first 19 prime bases up to 67. No K2-spsp’s < 1036 to the first 20 prime bases are found. Thus, the 100920 numbers prove Proposition 1.1. Table 1. List of all K2-spsp’s n = p (2p − 1) < 1036 to the first 18 prime bases spsp-base n = p (2p − 1) p 67 71 73 1543 26786 44434 20616 87767 76407 51301 27778299663977101 1 0 0 3573 96616 43156 88081 50098 90376 80081 42272722672638961 0 0 1 3957 57030 03519 44936 54580 63660 50221 44483537968286341 0 0 0 7434 11233 75303 11731 29413 76567 93141 60967664944338781 0 0 0 11068 90507 54608 90469 45172 83403 23501 74393901213274501 0 0 1 15837 76949 08904 27362 08053 96976 19981 88988115753988261 0 0 0 28474 73406 99486 58439 94849 77173 56181 119320438462881661 0 0 0 40367 25471 42188 26380 27978 16596 56101 142069093602758701 0 1 0 71361 96942 79138 04551 29896 91981 34381 188894109791589061 0 1 0 82824 36407 51608 91096 44501 71525 30181 203499833016099661 1 0 0 88957 81514 98055 54548 98662 46790 89781 210900231329656861 0 0 1 1 55559 41557 67059 71544 44226 02410 85421 278890135695676741 0 0 0 1 79574 77020 19886 65981 89846 22792 23261 299645432304572821 0 1 0 1 94375 57764 05156 30021 54227 05343 35741 311749561058644981 0 1 0 2 19337 01033 96078 90908 19416 68785 91181 331162354699026661 0 0 0 2 52100 58000 17252 74015 37676 90691 76741 355035617932711981 0 0 0 2 55173 55898 34500 77412 12085 09551 48501 357192916351549501 1 0 0 4 07387 35132 62980 70201 48352 55040 09841 451324357489321681 0 0 0 5 27946 43421 43471 71361 92967 10213 53301 513783239418311101 0 0 0 5 34925 55849 14817 98612 70909 62290 28021 517168037726366941 0 0 0 8 08460 12435 54390 22735 02512 76681 37421 635790895010080741 1 0 0 8 12968 03943 72552 74477 32866 69713 53901 637560992939991301 0 0 0 8 16345 93783 72388 10402 16654 72644 26261 638884159232813821 0 1 0 9 64006 87022 43616 26772 79703 83092 14301 694264672233998101 0 0 0 Define the function F (t, L)=#{N : N is a K2-spsp Table 2. The function F (t, L) t 9 10 11 12 13 14 15 16 17 18 19 L =1024 214 41 6 3 0 0 0 0 0 0 0 L =1028 15099 2680 551 105 12 1 0 0 0 0 0 L =1032 1146700 199736 38915 6913 1290 224 49 11 2 0 0 L =1036 90002828 15684487 3087051 546925 100920 18924 3778 664 128 24 4 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRONG PSEUDOPRIMES TO 1036 2101 36 12 3. C3-strong pseudoprimes < 10 with heights < 10 Thetriple(h1,h2,h3) is called C3-spsp(2)-acceptable if the hi are all odd positive integers, pairwise relatively prime, and h1 ≡ h2 ≡ h3 mod 4. Let q1 d ≡ x0 mod 4H, where x0 = x0 + j0H ≡ 2mod4,j0 =(2− x0)H mod 4, 0 ≤ j0 ≤ 3, x0 is the unique integer with 0 ≤ x0 (3.1) x0 ≡ −h − h mod h , ⎩⎪ 1,2 3,2 2 −h1,3 − h2,3 mod h3, −1 ≤ ≤ and hi,j = hi mod hj for 1 i = j 3. In [10, Section 2], we described a procedure for finding C3-spsp(2)’s, to a given limit, with heights bounded. The method is based on the following Lemma 3.1. Lemma 3.1 ([10, Theorem 2.1]). Let N = q1q2q3 be a product of three different odd primes. Then we have N is a C3-spsp(2) ⇐⇒ its kernel d is C3-spsp(2)-acceptable. Let bi be the ith prime, t ≥ 2andMt =4b2 ···bt, and suppose that (h1,h2,h3) is C3-spsp(2)-acceptable. For an odd prime b, define the set b b b (h1,h2,h3) ≤ Sb = u : u =2+4k, 0 k Thetriple(h1,h2,h3) is called C3-spsp(b1,b2,...,bt)-acceptable if the system of linear congruences x ≡ x mod 4H, (3.2) 0 x ≡ u mod 4b for some u ∈ S(h1,h2,h3), 2 ≤ i ≤ t, i i i bi has solutions, or in other words, the system x ≡ x mod 4H, (3.3) 0 ≡ ∈ (h1,h2,h3) x r mod Mt for some r Rt has solutions. The kernel d is called C3-spsp(b1,b2,...,bt)-acceptable if (h1,h2,h3) is C3-spsp(b1,b2,...,bt)-acceptable and (3.3) holds with x replaced by d. In [10, Section 4], we speeded up the procedure described in [10, Section 2] so 50 that we can find all C3-spsp’s less than a larger limit L =10 ,withthesame signature (1, 37, 41), to the first t ≥ 9 prime bases. The accelerated procedure is based on the following Lemma 3.2. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2102 ZHENXIANG ZHANG Lemma 3.2 ([10, Corollary 4.1]). Let N = q1q2q3 be a product of three different odd primes and let bi be the ith prime and t ≥ 2; and suppose (h1,h2,h3) is C3- spsp(2)-acceptable. Then N is a C3-spsp(b1,b2,...,bt) if and only if its kernel d is C3-spsp(b1,b2,...,bt)-acceptable. 36 Now our objective is to find all N = q1q2q3 and √ H0 H1 1 2 max h2, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRONG PSEUDOPRIMES TO 1036 2103 9 with H = h1h2h3 < 10 Do begin Using the Euclidean Algorithm and the Chinese Remainder Theorem, compute the seed x0 of the triple (h1,h2,h3); x0 ← x0; j0 ← (6 − x0 mod 4)H mod 4; If j0 > 0 Then x0 ← x0 + j0H; Compute the sets S(h1,h2,h3) for 2 ≤ i ≤ t; bi (h1,h2,h3) Using the Chinese Remainder Theorem, compute the set Rt ; (h1,h2,h3) For each element r of the set Rt Do Begin Using the Chinese Remainder Theorem, find a solution x x ≡ r mod Mt; For i := 1 To 3 Do qi ← xhi +1; q1q2 ← q1 · q2; N ← q1q2 · q3; If N (3.4) f(t, L, H)=#{N : N is a C3-spsp(b1,b2,...,bt) (h1,h2,h3) each R9 is of suitable size to fit in the memory of a PC Pentium IV/1.8GHz. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2104 ZHENXIANG ZHANG 36 Table 3. List of all C3-spsp’s n = q1q2q3 < 10 to the first 15 prime 12 bases with signature (h1,h2,h3) and height H =h1h2h3 <10 spsp-base n = q1q2q3 q1 h1 h2 h3 53 59 61 168 79087 75236 76911 80919 24541 71451 4674035851 1 29 57 0 1 0 404 00641 91965 17352 34018 69081 76331 8456744011 17 21 9193 0 0 1 638 44829 45542 74988 12832 64086 78791 885448831 1 29 31713 0 0 1 7386 49195 93228 96977 41258 18632 67131 6336781291 1 13 2233 1 0 0 14987 06615 24763 89521 37359 79490 57667 34754907427 1 17 21 0 0 0 22072 88430 42828 12126 78317 84005 82551 53394561991 1 5 29 0 0 1 22237 51676 18766 76325 87593 37145 05851 90866269051 5 13 57 0 1 0 22535 13758 55715 82393 57550 88590 41539 2423807299 1 341 4641 1 0 0 33197 80568 91606 40280 97700 07302 95751 42096868351 1 5 89 0 0 0 73956 31848 86374 65060 47922 10797 09511 4862377591 5 261 61621 0 0 1 84699 94086 53768 76461 36135 59745 27971 51195714571 3 23 247 0 1 0 86114 94622 43439 91256 73057 32901 32851 243674295091 41 69 145 0 1 1 1 57909 67938 52590 51550 19158 97075 64171 15045483979 1 165 281 0 0 0 1 64290 81802 11124 38861 65231 15234 05671 191971388071 3 11 19 0 0 1 3 29688 82238 37447 72420 05277 30066 57151 5187780511 1 5 472269 0 0 1 3 81295 37619 60634 49827 51838 84631 05491 122980397539 1 5 41 0 0 0 4 88694 76611 41627 89165 23631 96679 44651 112306837411 1 5 69 0 1 1 6 15953 28171 54420 32393 10712 67951 94451 326850934411 5 9 49 0 0 0 6 63871 52401 35144 81984 27536 68390 39183 187835202127 11 31 391 0 0 0 6 78033 04770 25874 16969 29260 30947 02251 114879489139 3 23 175 0 0 1 Table 4. The function f(t, L, 1012) t 9 10 11 12 13 14 15 16 L =1024 21 8 3 0 0 0 0 0 L =1028 217 70 20 6 1 0 0 0 L =1032 2808 821 249 70 17 6 0 0 L =1036 43278 12623 3655 1019 271 80 20 2 Table 5. The function f(t, 1036, H) t 9 10 11 12 13 14 15 16 H =102 894 211 51 16 4 1 0 0 H =103 11592 3225 814 221 56 15 6 0 H =104 22249 6367 1717 482 125 30 9 0 H =105 30385 8811 2531 710 176 42 13 1 H =106 35526 10377 2995 846 216 57 16 1 H =107 38811 11344 3287 922 238 69 19 2 H =108 40872 11915 3454 966 253 76 20 2 H =109 42070 12270 3548 989 264 79 20 2 H =1010 42731 12469 3609 1003 268 80 20 2 H =1011 43087 12567 3637 1011 270 80 20 2 H =1012 43278 12623 3655 1019 271 80 20 2 4. Discussion H In Tables 6, 7, 8, 9, and 10 we give f(t, ψu, )foru = 13, 14, 15, 16 and 18; and ≥ H≤ 12 H H 12 ≥ for t 9, 10 . In these tables, if f(t, ψu, )=0for =10 and for t t0, then the columns for t ≥ t0 are all deleted since all entries are 0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRONG PSEUDOPRIMES TO 1036 2105 Table 6. H The function f(t, ψ13, ) t 9 10 11 H =103 6 3 2 H =104 19 11 4 H =105 24 12 5 H =106 25 12 5 H =107 26 12 5 H =108 27 13 5 H =109 28 13 5 H =1010 28 13 5 H =1011 28 13 5 H =1012 28 13 5 Table 7. H The function f(t, ψ14, ) t 9 10 11 12 13 H =102 4 1 0 0 0 H =103 49 15 3 0 0 H =104 95 37 9 1 0 H =105 129 44 12 2 0 H =106 150 52 13 2 0 H =107 161 56 14 2 0 H =108 172 58 14 2 0 H =109 179 60 15 3 1 H =1010 180 60 15 3 1 H =1011 180 60 15 3 1 H =1012 182 61 15 3 1 Table 8. H The function f(t, ψ15, ) t 9 10 11 12 13 H =102 9 4 0 0 0 H =103 91 26 5 1 0 H =104 180 60 15 2 0 H =105 245 71 21 5 0 H =106 283 83 25 5 0 H =107 311 93 26 5 0 H =108 336 97 27 5 0 H =109 344 99 28 6 1 H =1010 348 101 28 6 1 H =1011 348 101 28 6 1 H =1012 350 102 28 6 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2106 ZHENXIANG ZHANG H H 12 From Table 6 we may predict that f(13,ψ13, )=0for > 10 and so that (4.1) ψ13 <ψ13. H H 12 From Table 7 we may predict that f(14,ψ14, )=0for > 10 and so that (4.2) ψ14 <ψ14. H H 12 From Table 8 we may predict that f(15,ψ15, )=0for > 10 and so that (4.3) ψ15 <ψ15. H H 12 From Table 9 we may predict that f(16,ψ16, )=0for > 10 and so that ≤ (4.4) ψ16 = ψ17 <ψ16 ψ17. H H 12 From Table 10 we may predict that f(18,ψ18, )=0for > 10 and so that ≤ (4.5) ψ18 = ψ19 <ψ18 ψ19. Table 9. H The function f(t, ψ16, ) t 9 10 11 12 13 H =102 11 4 0 0 0 H =103 165 50 13 4 1 H =104 332 106 30 7 2 H =105 452 128 42 12 2 H =106 519 146 46 12 2 H =107 570 164 50 14 2 H =108 605 169 51 14 2 H =109 618 175 52 15 3 H =1010 622 177 52 15 3 H =1011 625 177 52 15 3 H =1012 630 179 53 15 3 Table 10. H The function f(t, ψ18, ) t 9 10 11 12 13 14 15 H =102 133 31 8 3 0 0 0 H =103 1641 438 104 29 6 0 0 H =104 3212 900 251 75 19 4 1 H =105 4424 1242 370 110 26 6 1 H =106 5207 1505 449 133 30 9 2 H =107 5714 1658 491 142 33 11 3 H =108 6036 1745 519 150 35 13 3 H =109 6211 1797 527 153 38 14 3 H =1010 6309 1832 544 157 39 15 3 H =1011 6352 1842 547 158 39 15 3 H =1012 6384 1853 549 159 39 15 3 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRONG PSEUDOPRIMES TO 1036 2107 Since the conditions for a number to be a C3-spsp are more stringent than those for it to be a K2-spsp, K2-spsp’s are much more numerous than C3-spsp’s as can be seen from Tables 2 and 4. Thus, combining (4.1)-(4.5) and (1.8), it is reasonable to predict that (4.6) ψt <ψt for any t ≥ 12. Acknowledgment I thank the referee for kind and helpful comments that improved the presentation of the paper. References 1. R. Crandall and C. 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Comp. 72 (2003), 2085–2097. http://www.ams.org/journal-getitem?pii=S0025-5718- 03-01545-X MR1986825 (2004c:11008) Department of Mathematics, Anhui Normal University, 241000 Wuhu, Anhui, People’s Republic of China E-mail address: [email protected] E-mail address: ahnu [email protected] URL: http://www.ahnu.edu.cn/~math/jiaoshou/zzx.htm URL: http://my.hn8868.com/zhangzhx/ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use