source: http://web.archive.org/web/19990506090117/http://www.utm.edu/research/primes/notes/errata.html

Special thanks to C .R. J. Currie, Marc Deléglise, Harvey Dubner, Bill Dubuque, Tony Forbes, Warut Roonguthai, Jörg Richstein, Carlos B. Rivera, and Luiz Rodriguez Torres, for pointing out errors in Ribenboim's book (and sometimes in this page of errata!)

- Chapters:
- How Many Primes Are There? (pp.1-18)
- How to Recognize Whether a Natural Number is Prime (pp. 19-178)
- Are There Functions Defining Prime Numbers? (pp. 179-212)
- How Are the Prime Numbers Distributed? (pp. 213-322)
- Which Special Kinds of Primes Have Been Considered? (pp. 323-370)
- Heuristics and Probablistic Results about Prime Numbers (pp. 371-426)

- Supplements:
- Conclusion
- Bibliography
- The Pages That Couldn't Wait
- Primes up to 10,000
- Index of Tables
- Index of Names
- Subject Index

- VIII.
**Euclidean Sequences (pp. 11-16)**

- IV.
**Lucas Sequences (pp. 54-74)**- 56
**Typo**: Carlos B. Rivera notes that the second formula in (IV.3) should read*V*_{m+n}= V_{m}V_{n}– Q^{n}V_{m-n}= ...

- 56
**Mersenne Numbers (pp. 90-103)**- 90
**Error**: The second paragraph begins "M_{q}= 2^{q}– 1 (with*p*prime)." Clearly the*p*should be*q*. - 91
**Error**: The theorem which readsIf

needs the added assumption "Let*n*divides M_{q}, then*n*= ±1 (mod 8) and*n*= 1 (mod*q*)*q*be an odd prime." Otherwise we have the counterexample*n*=3 which divides M_{2}. - 91
**Error**: The second sentence of the proof has "2^{q}=1 (mod*q*)" when it means "2^{q}=1 (mod*p*)". - 96
**Error**: The number 47094312^{.}2^{16352}– 1 should be 47094312**9**^{.}2^{16352}– 1. Also, this prime, as well as 157324389^{.}2^{16352}– 1 were found by Indlekofer and Járai (not Dubner). [These errors were noted by Warut Roonguthai.] - 96
**Error**: A larger composite Mersenne than the one he lists as the record is M_{q}, where*q*= 8069496435^{.}10^{5072}– 1 (a Sophie Germain prime mentioned on page 330). [Again noted by Warut Roonguthai.]

- 90
**Curious Primes (pp. 159-162)**- 161
**Error**: Record A reads 72323252323272325252(10^{3120}– 1)/(10^{20}– 1) but this number is obviously even! Add one to get the actual prime. - 162
**Error**: Part I reads "The smallest prime with 1000 digits is 100^{999}+212^{.}10^{499}+1." This should read "smallest**known**prime." The smallest is almost certainly 10^{999}+7 (a probable-prime). - 162
**Error**: The same goes for the next two primes listed in part I. The prime number theorem shows they are not the least such examples.

- 161

**Prime-Producing Higher-Degree Polynomials (p. 203)**- 203
**Error**: "assume prime values at 0, 1, ..., 2.5." The last number is 25 (not 2.5). (Both polynomials give composite values at –1 and 26.)

- 203
**Races for Quadratic Polynomials (p. 204-206)**- 205
**Error**: The definition of P_{0}[*f*(*X*),*N*] is in error. Since we are interested in the smallest prime factor the definition should readP

(that is, change max to min). Next, Rodriguez Torres points out that if we let_{0}[*f*(*X*),*N*] = min { P_{0}[*f*(*k*)] |*k*= 0, 1, 2, ...,*N*}.*m*=P_{0}[*f*(*X*),*N*] and*m*<*N*, then we must have*m*=P_{0}[*f*(*X*),*L*] for all*L*>*m*. So a better definition might beP

_{0}[*f*(*X*)] = min { P_{0}[*f*(*k*)] |*k*= 0, 1, 2, ... }. - 205
**Error:**Luiz Rodriguez Torres notes that the records listed at the bottom of page 205 and top of 206 are for quadratics of the form*X*^{2}+*X*+*n*where*n*is prime. If we drop the primality condition on*n*, then the record is 181 for*X*^{2}+*X*+132874279528931 found in 1990 by Fung & Williams. - 205
**Addition:**In July 1996 Rodriguez Torres set a new record (for prime constants) with P_{0}[*X*^{2}+*X*+67374467] = 107. - 206
**Error:**First line should read "Previously he found P_{0}[*X*^{2}+*X*+601037] = 61." Rodriguez Torres notes that this was his error.

- 205

**The growth of π(***x*) (pp. 215-248)- 222
**Error**: In Mertens formula, on the bottom of the page, log*n*should be replaced by log*p*. See [HW79, theorem 429]._{n} - 238
**Error**: In Table 27 the entry for π(3.10^{17}) should be7 650 011 911 220 803

(not 7 650 011 911 275 069). On 10/23/96 Marc Deléglise recalculated and verified this value (which matches the value in their article [DR96, p244]). Ribenboim's values for Li(*x*) – π(*x*) and R(*x*) – π(*x*) are also in error for 3.10^{17}. (These errors*may*have come from typos in preprints of the paper [DR96].) - 236,238
**Addition**: Deléglise, working alone, has extended these results as follows:π(2e18) = 48 645 161 281 738 535 π(3e18) = 72 254 704 797 687 083 π(4e18) = 95 676 260 903 887 607 π(4185296581467695669) = 100 000 000 000 000 000 π(5e18) = 118 959 989 688 273 472 π(6e18) = 142 135 049 412 622 144 π(7e18) = 165 220 513 980 969 424 π(8e18) = 188 229 829 247 429 504 π(9e18) = 211 172 979 243 258 278 π(1e19) = 234 057 667 276 344 607 π(2e19) = 460 637 655 126 005 490 π(4e19) = 906 790 515 105 576 571 π(1e20) = 2 220 819 602 560 918 840

- 222
**Gaps Between Primes (pp. 250-259)**- 251
**Addition:**Text notes Shanks conjectured log*p*(*g*) is apporximately sqrt(*g*). Luis Rodriguez Torres notes [email 12 Jun 1996] "Based in convincing probabilistic considerations, its better [to write]:

It perfectly fits with the last data available."*g*= (log*p*- log log*p*)^{2}.

- 251
**Twin Primes (pp. 259-264)**- 263
**Errors**: Jörg Richstein noted the following typos- 2nd and 3rd line:
- Kutnib should be Kutrib
- D. Richstein should be J. Richstein

- List of historical counts of Twin Primes:
- Glaishir should be Glaisher
- Lehmir should be Lehmer
- Armendiny should be Armerding
- "Brent (1995)" should be "Brent (1975)"

- 2nd and 3rd line:
- 263-264
**Addition:**Far larger twin primes (with more than*twice*the digits) have now been found. See the Largest Known Primes page.

- 263
**Primes in Arithmetic Progression (pp. 265-287)**- 287
**Error:**The last three lines of Table 32 contain several errors. They should read:20 214861583621 18846497670 572944039351

Only one of these numbers were "wrong," six were just in the wrong place. (Error first noted by Rodriguez Torres.)*s*F Mar 1987 21 5749146449311 26004868890 6269243827111*s*P 1992 22 11410337850553 4609098694200 108201410428753 P,M,T 1995

- 287
**Distribution of Carmichael Numbers (pp. 314-317)**- 316
**Error:**End of paragraph two: "by Ketter in 1988" should be "by Keller in 1988." (Error noted by Dubner.)

- 316

**Sophie Germain Primes (pp. 329-333)**- 331
**Error:**Dubner's palindromic Sophie Germain primes should be*p*= 39493939493 and 2*p*+1 = 78987878987. (Error noted by Rodriguez Torres, corrected by Dubner).

- 331
**Addendum on Cullen Numbers (pp. 360-361)**- 360
**Error**: C_{n}is prime for*n*=6611 (not 5611). (Error noted by Keller.) - 360
**Error**: (Third paragraph) "Dubner and Riesel studied divisibility properties..." should be "Dubner and*Keller*studied..." (Error noted by Dubner.) - 361
**Error**: C'_{n}is prime for*n*=822 (not 882). (Error noted by Keller.) - 361
**Addition**: Keller's 1995 paper referred to has appeared:**Wilfrid Keller**, "New Cullen primes,"*Math. Comp.,***64**(1995) 1733-1741; Supplement, ibid., S39-S46.

- 360

**Bibliography (pp. 433-507)**- 494
**Error**: The pages numbers for "1993 Pinch" should be 381-391 not 703-722. (Error noted by Dubner.) - 494
**Addition**: "1994 Deléglise and Rivat" has appeared (and has been extended as noted above):**M. Deléglise**and**J. Rivat,**"Computing pi(*x*): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method,"*Math. Comp.,***65**(1996) 235-245.

- 494

**Addendum "The Pages That Couldn't Wait" (pp. 509-511)**- 511
**Error**: The 13-tuple beginning*p*= 28561589689237439 and the 14-tuple beginning*p*= 79287805466244209 were discovered by Dimitrios Betsis & Sten Sfholm, 1982 (not Forbes). (Error noted by T. Forbes.) - 510
**Addition:**Warut Roonguthai says "Here's the result of my search for the smallest*n*-digit prime*p*= 10^{n-1}+*k*such that*p*+2,*p*+6, and*p*+8 are also prime, i.e. (*p*,*p*+2,*p*+6,*p*+8) is the smallest*n*-digit prime quadruplet:**Smallest***n*-digit Prime Quadruplets*n**k**n**k*100 349,781,731 400 34,993,836,001 200 21,156,403,891 500 883,750,143,961 300 140,159,459,341 In none of these cases is

*p*+12 or*p*-4 a prime." See his email. - 510
**Addition:**Prime triplets, with 1083 digits,*N*-5,*N*-1,*N*+1, with*N*= 437850590*(2^{3567}- 2^{1189}) - 6*2^{1189}, Tony Forbes, 1997. - 510
**Addition:**Prime 6-tuplets, with 155 digits,*p*,*p*+4,*p*+6,*p*+10,*p*+12,*p*+16, with*p*= 2^{512}+ 6638977280721, Tony Forbes, 1997. - 511
**Addition:**Prime 10-tuplets, with 40 digits,*p*,*p*+2,*p*+6,*p*+12,*p*+14,*p*+20,*p*+24,*p*+26,*p*+30,*p*+32, with 22^{.}10^{38}+2241278889512317 Tony Forbes, 1997. - 511
**Addition:**13-tuplets, with 19 digits,*p*,*p*+2,*p*+8,*p*+14,*p*+18,*p*+20,*p*+24,*p*+30,*p*+32,*p*+38,*p*+42,*p*+44,*p*+48, where*p*= 3356052825826535669, Tony Forbes, 1995. (Noted by T. Forbes). - 511
**Addition:**Prime 14-tuplets, with 19 digits,*p*,*p*+2,*p*+8,*p*+14,*p*+18,*p*+20,*p*+24,*p*+30,*p*+32,*p*+38,*p*+42,*p*+44,*p*+48,*p*+50, where*p*= 6120794469172998449.

- 511

**Index of names (pp. 519-534)**- 520
**Error**: Bichstein should be Richstein. (Error noted by J. Richstein.)

- 520

Another Prime Page by Chris K. Caldwell