This article was printed in the Journal of Recreational Mathematics, Vol 17:4, 1984-85.

SINKERS OF THE TITANICS

SAMUEL YATES

On September 5, 1984, Professor Wilfrid Keller of the University of Hamburg proved that the 7,067-digit number 5 · 2^{23473} + 1 is prime. It is the largest non-Mersenne prime known, and it is larger than all known Mersenne primes except for three of them. In 1971, Bryant Tuckerman discovered a 6,0002-digit Mersenne prime [1], and, although five larger Mersenne primes have been found since then, Professor Keller’s discovery is the only known non-Mersenne prime that is larger than Tuckerman’s prime, but it may not enjoy this unique status very long.

On September 4, one day before Keller’s find, Harvey Dubner of Dubner Computer Systems in Fort Lee, New Jersey, believed that he had found the largest known non-Mersenne prime when he observed that 4974 · 10^{4796} + 1 is prime. It is a 4,800-digit number, writed as 4974 followed by 4,795 zeros, and a 1. Actually, Professor Keller had found a 5,573-digit prime in mid-August, but news of the discovery had not been reported abroad immediately. Sic transit gloria!

In a recent paper [2], primes that are written with a thousand or more decimal digits were called titanic primes. A total of 319 of them were listed in an accompanying table; six more were appended before publication. Before 1979, only seven titanic primes were known, and all of them were Mersenne primes.

In the past year, there has been a barrage of newly found primes that have sunk most of the listed ones deeper down the nether regions of the table. By January 1, 1985, the number of known titanic primes had increased to 581, and the number with more than 2,000 digits had gone from 73 to 170.

The sinkers of the titanics are some of the same people who had made many of the earlier discoveries. Besides Keller and Dubner, they are Hiromi Suyama of Karatsu, Japan, and Professor A. Oliver L. Atkin of the University of Illinois at Chicago Circle. All of them use techniques that are modifications and improvements of methods which have been described in recent journals. Because they constantly make changes and are eager to see what kinds of results they can obtain, their production methods and the quality of their output are more advanced that what appears in current publications. It is by an interchange of correspondence with these men that I have been fortunate enough to serve somewhat as a receiving and disbursing agent for news of their accomplishments.

It is quite appropriate to compare them and other producers of large primes with those men who once discovered and explored new lands and pioneered in establishing new settlements. They exemplify adventurousness, imagination, solid background knowledge, perseverance, and patience. Despite this analogy, there is no stereotype, each individual achieving significantly in his own way.

Date: 1985, Baywood Publishing Co, Inc.

Professor Atkin has been working with Neil Rickert, generating primes that are mostly of the form k · 2^{n} + 1, in their successful quest to obtain large twin and non-Mersenne primes, as well as factors of Fermat numbers. Theirs was for a while the largest known non-Mersenne prime. Their 2,003-digit twin primes 520995090 · 2^{6624} ± 1 are the largest twins known. They have also found titanic twin primes 219649815 · 2^{4481} ± 1 (1,358 digits) and 256200045 · 2^{3426} ± 1 (1,040 digits). A large number of primes with more than 2,000 digits were found by them. Professor Atkin recently wrote, ”We have an IBM 3081D now, bigger than before, but without any of the special number-crunching features of the Cray or the British DAP machine. We are inclined to go, when we have time, for some prime triplets (b, b + 2, b + 6) since there is some skill and knowledge involved in proving all three prime, and one is less handicapped by a smaller machine.

Most of Harvey Dubner’s number theoretic computations have been with numbers related to repunits (numbers R(n) written as strings of n 1’s) and powers of 10. He supplied many large primitive divisors of repunits for tabulations maintained by this writer as well as by Professor Sam Wagstaff of the “Cunningham Project.” He has extended the work of H.C. Williams of the University of Manitoba so that it can now be said that all repunits above R(1031) and below R(6197) are composite. Recently Professor Atkin contributed to Dubner’s discovery of large twin primes by verifying that one of each of two twin pairs was prime.

Professor Williams generated large primes containing long repdigit strings, including some primes with long repunit strings [3]. These are among the special types of large primes that Rudolf Ondrejka collects. Spurred on by Ondrejka, who writes about these fascinating numbers, Dubner has produced many of them. The reader may verify the interesting fact that Dubner’s primes in Table 1, designated as palidromes are indeed true palindromes. Most of Dubner’s titanic primes contain large repdigits. His 2,188-digit 872! + 1 and 4,042-digit 1477! + 1 and numbers of the form 2 · 3 · 5 · 7 … p + 1 (i.e., 1 more than the product of the ascending primes sequence), where p = 4787, 4547, and 3229 are exceptions. He has found that numbers of this form are composite when p is greater than 4787 and less than 10133. One other titanic prime that he found when Ondrejka expressed interest in that type is the 1,201-digit number written as seven successive strins of 12345667890 followed by 1,1130 zeroes and a 1. Since then, he has discovered even larger pandigital numbers, the largest of which are shown in Table 1.

Although the computer that Dubner uses is based on an INTEL 8080 microprocessor similar to common home computers, it has been so modified by him and his son that he feels that when used for number theory, it “is within a factor (in speed) of five or ten of a multi-million dollar Cray computer. It appears to be most economical, since its total hardware cost of operation, including amortization, is about $10.00 per day.” He writes that “I can find about two 1,000-digit primes per hour. I discovered the 4,074-digit prime in about 12 hours. It took about six days to discover the 4,800-digit prime.”

Hiromi Suyama is a diligent collector and producer of large primes who has researched and written extensively on the subject of factors of Fermat numbers. He writes, ”My computers were an 8-bit microprocessor Z-80 (2MHz) and an 8bit microprocessor MC6809(1MHz). I am now using the MC6809 and a 16-bit microprocessor iAPX 86/20 (=8086 + co-processor 8087, about 4.9 MHz) which is a CPU in a home computer PC-9801 (like IBM-PC). It took 1 hour, 8 minutes and 11 seconds to prove that the 1,694-digit number 53 · 2^{5621} + 1 is prime.”

Table 1. The Largest Known Primes

A(K, N) = K^{⋆}(2^{⋆⋆}N) + 1  B(K, N) = K^{⋆}(2^{⋆⋆}N) − 1
E(K, N) = (K^{⋆⋆}2)^{⋆}(2^{⋆⋆}N) + 1  F (K, N) = K^{⋆⋆}4)^{⋆}(2^{⋆⋆}N) + 1
G(K, N) = (K^{⋆}(10^{⋆⋆}N) + 1  R(N) = (10^{⋆⋆}N − 1)/9

P (N, K, M) = (10^{⋆⋆}N + K)^{⋆}(10^{⋆⋆}M) + 1

No. Prime  Digits Discoverer^{a} Year Special Type
1 B(1,132049)  39751  S  1983 Mersenne
2 B(1,86243)  25962  S  1982 Mersenne
3 B(1,44497)  13395  SN  1979 Mersenne
4 A(5,23473)  7067  K  1984
5 B(1,23209)  6987  N  1979 Mersenne
6 B(1,21701  6533  NN  1978 Mersenne
7 B(1,19937)  6002  T  1971 Mersenne
8 A(18496,18496)  5573  K  1984 Cullen
9 G(7113,4897)  4901  D  1984
10 G(279,4898)  4901  D  1984
11 G(4974,4796)  4800  D  1984
12 A(7,15494  4666  K  1984
13 G(138,4071)  4074  D  1984
14 A(7,13496)  4064  K  1984
15 1477! + 1  4042  D  1984 “Factorial”
16 A(15450435,13281) 4006  AR  1983
17 A(4549545,13281)  4005  AR  1983
18 A(5,13165)  3964  K  1979
19 E(6486,12674)  3823  AR  1983
20 B(3,12676)  3817  BB  1979
21 A(139,12614)  3800  K  1979
22 B(9,12495)  3763  BR  1981
23 B(12379,12379)  3731  K  1984 Cullen
24 G(8166,3610)  3614  D  1984
25 A(19,11890)  3581  K  1980
26 G(378,3510)  3513  D  1984
27 B(9,11547)  3477  BB  1979
28 G(639,3410)  3414  D  1984
29 A(70195125,11202) 3380  AR  1980
30 B(1,11213)  3376  G  1963 Mersenne
31 G(3936,3310)  3314  D  1984

a Key to Discoverers: AR = A.Oliver L.Atkin, Neil W.Rickert; BB = Walter Borho, Jurgen Buhl; BR = Walter Borho, R.Reckow; CW = G.V.Cormack, Hugh C.Williams; D = Harvey Dubner; G = Donald B.Gillies; K = Wilfred Keller; N = Curt L.Noll; NN = Curt L.Noll, Laura A.Nickel; S = David Slowinski; SN = David Slowinski, Harry L.Nelson; SU = Hiromi Suymama; T = Bryant Tuckerman.

Table 1. (Cont’d.)

No. Prime  Digits Discoverer^{a} Year Special Type
32 G(2439,3310)  3314  D  1984
33 G(1053,3310)  3314  D  1984
34 A(111,10883)  3279  K  1981
35 G(6^{∗}R(1611),1612) 3223  D  1984
36 G(2373,3210)  3214  D  1984
37 G(357,3100)  3103  D  1984
38 G(3441,3080)  3084  D  1984
39 G(6834,3070)  3074  D  1984
40 G(4419,3070)  3074  D  1984
41 G(4095,3070)  3074  D  1984
42 A(167,10183)  3068  K  1979
43 A(11,10179)  3066  K  1979
44 G(1965,3020)  3024  D  1984
45 F(6952,9952)  3012  AR  1983
46 F(5555,9952)  3011  AR  1983
47 F(5213,9952)  3011  AR  1983
48 F(4682,9952)  3011  AR  1983
49 F(4638,9952)  3011  AR  1983
50 F(3950,9952)  3011  AR  1983
51 F(2081,9952)  3010  AR  1983
52 A(103858755,9952) 3004  AR  1980
53 A(31336305,9921)  2995  AR  1980
54 B(1,9941)  2995  G  1963 Mersenne
55 E(694,9920)  2992  AR  1980
56 P(1488,3,1488)  2977  D  1984 Palindrome
57 A(2897,9715)  2928  K  1979
58 B(1,9689)  2917  G  1963 Mersenne
59 G(2^{∗}R(1439),1440) 2879  D  1984
60 B(9531,9531)  2874  K  1984 Cullen
61 A(25,9522)  2868  K  1983
62 A(19,9450)  2847  K  1983
63 A(9,9431)  2840  K  1983
64 P(1419,5,1419)  2939  D  1984 Palindrome
65 P(1375,999,1373)  2749  D  1984 Palindrome
66 A(31,9096)  2740  K  1983
37 A(65057,8899)  2684  K  1983

a Key to Discoverers: AR = A.Oliver L.Atkin, Neil W.Rickert; BB = Walter Borho, Jurgen Buhl; BR = Walter Borho, R.Reckow; CW = G.V.Cormack, Hugh C.Williams; D = Harvey Dubner; G = Donald B.Gillies; K = Wilfred Keller; N = Curt L.Noll; NN = Curt L.Noll, Laura A.Nickel; S = David Slowinski; SN = David Slowinski, Harry L.Nelson; SU = Hiromi Suymama; T = Bryant Tuckerman.

Table 1. (Cont’d.)

No. Prime  Digits Discoverer^{a} Year Special Type
68 P(1335,6,1335)  2671  D  1984 Palindrome
69 G(6,2629)  2630  D  1984
70 G(3,2620)  2621  D  1984
71 P(1305,23456789198765432,1289) 2595  D  1984 Pandigital Palindrome
72 A(41,9411)  2534  K  1983
73 A(14899,8234)  2483  K  1983
74 P(1242,23456789198765432,1226) 2469  D  1984 Pandigital Palindrome
75 B(9,8007)  2412  BB  1980
76 A(19,7998)  2409  K  1983
77 A(9,7967)  2400  AR  1979
78 B(9,7939)  2391  BB  1980
79 A(29,7927)  2388  CW  1979
80 A(271,7780)  2345  K  1983
81 B(7755,7755)  2339  K  1984 Cullen
82 A(27,7639)  2301  CW  1979
83 A(41,7607)  2292  K  1983
84 A(39,7583)  2285  K  1983
85 B(3,7559)  2276  BB  1980
86 A(19,7498)  2259  K  1983
87 A(49,7446)  2244  K  1983
88 A(15,7392)  2227  K  1983
89 P(1118,56789123432198765,1102) 2221  D  1984 Pandigital Palindrome
90 P(1110,88088,1106)  2217  D  1984 Palindrome
91 A(17,7311)  2203  CW  1979
92 872! + 1  2188  D  1983 “Factorial”
93 G(R(820),1506)  2112  D  1984
94 A(15,7050)  2124  K  1983
95 G(5^{∗}R(606),1506)  2112  D  1984
96 A(21,6981)  2103  K  1983
97 A(9,6937)  2090  K  1983
98 A(45,6923)  2086  K  1983
99 A(73252,6889  2079  K  1983
100 A(19,6838)  2060  K  1983
101 A(15,6804)  2050  K  1983
102 2^{∗}3^{∗}6^{∗}7^{∗} …^{∗} 4787 + 1  2038  D  1984 “Prime-Factorial”
103 A(21,6712)  2022  K  1983

a Key to Discoverers: AR = A.Oliver L.Atkin, Neil W.Rickert; BB = Walter Borho, Jurgen Buhl; BR = Walter Borho, R.Reckow; CW = G.V.Cormack, Hugh C.Williams; D = Harvey Dubner; G = Donald B.Gillies; K = Wilfred Keller; N = Curt L.Noll; NN = Curt L.Noll, Laura A.Nickel; S = David Slowinski; SN = David Slowinski, Harry L.Nelson; SU = Hiromi Suymama; T = Bryant Tuckerman.

Table 1. (Cont’d.)

No. Prime  Digits Discoverer^{a} Year Special Type
104 G(6,1919)^{∗}12∗R(995) + 1 2015  D  1984
105 A(91,6668)  2010  SU  1984
106 P(1003,818,1001)  2005  D  1984 Palindrome
107 A(524477415,6624)  2003  AR  1984
108 A(520995090,6624)  2003  AR  1984 Twin
109 B(520995090,6624)  2003  AR  1984 Twin
110 A(517377630,6624)  2003  AR  1984

a Key to Discoverers: AR = A.Oliver L.Atkin, Neil W.Rickert; BB = Walter Borho, Jurgen Buhl; BR = Walter Borho, R.Reckow; CW = G.V.Cormack, Hugh C.Williams; D = Harvey Dubner; G = Donald B.Gillies; K = Wilfred Keller; N = Curt L.Noll; NN = Curt L.Noll, Laura A.Nickel; S = David Slowinski; SN = David Slowinski, Harry L.Nelson; SU = Hiromi Suymama; T = Bryant Tuckerman.

A great numbr of Keller’s titanic primes are of the forms k·2^{n}+1 and k(2^{n}−1)+1. He has factored Fermat numbers and found large twin primes. He writes, “Investigating the so-called Cullen numbers C_{n} = n · 2^{n} + 1 and W_{n} = n · 2^{n} − 1, I established that C_{n} is prime for n = 4713, 5795, 6611 (previously only n = 141 was known to give a prime), and W_{n} is prime for n =2,3,6,30,75,81,115, 249,362,384,462,512,751,882,5312,7755,9531 (including some previously known primes, like the Mersenne number m_{521} = W_{512}.” His work was done on a TELEFUNKEN TR440 computer and a SIEMENS 7.882 computer, and the methods and techniques that he used until 1983 were described by him in Mathematics of Computation [4].

Table 1 shows the 110 largest known titanic primes, updated to January 1, 1985.

Added in Proof - and updating to April 1, 1985.

Harvey Dubner discovered a new largest non-Mersenne prime! It is the 7,094digit number 6006×10^{7090}+1. It required two days of running time on his computer. He also found a few probable twin primes larger than the listed known twins. While doing so, he generated more primes with more than 2,000 digits. As of April 1, the total number of titanic primes is 652. Dubner’s present largest “pandigital prime” is the 3,284-digit number

1_{111}2_{111}3_{111}4_{111}5_{111}6_{111}7_{111}8_{111}9_{111} × 10^{2285} + 1,

using a chemical-type notation.

While proving that the numbers tested are all composite, he has extended the search for additional primes among numbers of the form n! + 1 to n = 2043; among numbers of the form 2 × 3 × 5 × ×7 ⋯ × p + 1 to p = 11213; and among repunits to R(6828).

No new Mersenne primes have been found since 1983, and no new prime repunits have been verified in almost a decade since R(317) was discovered. Efforts are still being made to show that probable prime R(1031) is prime. The prime that is shown in Table 1 as the 110th largest is now the 147th largest known prime.

References

1. B. Tuckerman, The 24th Mersenne Prime, Proceedings of the National Academy of Science USA, 68, pp.2319-2320, 1971.

2. S. Yates, Titanic Primes, Journal of Recreational Mathematics, 16:4, pp.250-262, 1983-84.

3. H. C. Williams, Some Primes with Interesting Digit Patterns, Mathematics of Computation, 32, pp. 1306-1310, 1978.

4. W. Keller, Factors of Fermat Numbers and Large Primes of the Form k · 2^{n} + 1, Mathematics of Computation, 32:144, pp.661-673, October 1983.

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