Notes with green background are already inserted into the book. — E.G.A.
The New Book of Prime Number Records Additions and Errata
We have found Paulo Ribenboim's text The New Book of Prime Number Records, 3ed. (Springer-Verlag New York, 1995; QA246 .R47; ISBN 0-387-94457-5) to be an invaluable reference and recommend it to everyone interested in primes. To make this book even more useful we offer the following list of errors and additions. If you know of others please let us know. (Note: Ribenboim has named this the "official" site for the collections of typos and errata from his text--all will be forwarded to him.)
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!)
16Error: In the list of primes of the form p# – 1, the prime 13# – 1 is omitted. This is my error! Ribenboim took his list from [CD93] and [CD95] which both omit this prime—sorry.
Chapter 2: How to Recognize Whether a Natural Number is Prime
IV. Lucas Sequences (pp. 54-74)
56Typo: Carlos B. Rivera notes that the second formula in (IV.3) should read
Vm+n = VmVn – QnVm-n = ...
Mersenne Numbers (pp. 90-103)
90Error: The second paragraph begins "Mq = 2q – 1 (with p prime)." Clearly the p should be q.
91Error: The theorem which reads
If n divides Mq, then n = ±1 (mod 8) and n = 1 (mod q)
needs the added assumption "Let q be an odd prime." Otherwise we have the counterexample n=3 which divides M2.
91Error: The second sentence of the proof has "2q=1 (mod q)" when it means "2q=1 (mod p)".
96Error: The number 47094312.216352 – 1 should be 470943129.216352 – 1. Also, this prime, as well as 157324389.216352 – 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 Mq, where q = 8069496435.105072 – 1 (a Sophie Germain prime mentioned on page 330). [Again noted by Warut Roonguthai.]
Curious Primes (pp. 159-162)
161Error: Record A reads 72323252323272325252(103120 – 1)/(1020 – 1) but this number is obviously even! Add one to get the actual prime.
162Error: Part I reads "The smallest prime with 1000 digits is 100999+212.10499+1." This should read "smallest known prime." The smallest is almost certainly 10999+7 (a probable-prime).
162Error: The same goes for the next two primes listed in part I. The prime number theorem shows they are not the least such examples.
Chapter 3: Are There Functions Defining Prime Numbers?
203Error: "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.)
Races for Quadratic Polynomials (p. 204-206)
205Error: The definition of P0[f(X),N] is in error. Since we are interested in the smallest prime factor the definition should read
P0[f(X),N] = min { P0[f(k)] | k = 0, 1, 2, ..., N }.
(that is, change max to min). Next, Rodriguez Torres points out that if we let
m=P0[f(X),N] and
m<N, then we must have
m=P0[f(X),L] for all
L>m. So a better definition might be
P0[f(X)] = min { P0[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
X2+X+n where n is prime. If we drop the primality condition on n, then the record is 181 for
X2+X+132874279528931 found in 1990 by Fung & Williams.
205Addition: In July 1996 Rodriguez Torres set a new record (for prime constants) with P0[X2+X+67374467] = 107.
206Error: First line should read "Previously he found P0[X2+X+601037] = 61." Rodriguez Torres notes that this was his error.
Chapter 4: How Are the Prime Numbers Distributed?
The growth of π(x) (pp. 215-248)
222Error: In Mertens formula, on the bottom of the page, log n should be replaced by log pn. See [HW79, theorem 429].
238Error: In Table 27 the entry for π(3.1017) should be
7 650 011 911 220 803
(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 3.1017. (These errors may have come from typos in preprints of the paper [DR96].)
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]:
g = (log p - log log p)2.
It perfectly fits with the last data available."
Twin Primes (pp. 259-264)
263Errors: 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)"
263-264 Addition: Far larger twin primes (with more than
twice the digits) have now been found. See the Largest Known Primes page.
Primes in Arithmetic Progression (pp. 265-287)
287Error: The last three lines of Table 32 contain several errors. They should read:
20 214861583621 18846497670 572944039351 s F Mar 1987
21 5749146449311 26004868890 6269243827111 s P 1992
22 11410337850553 4609098694200 108201410428753 P,M,T 1995
Only one of these numbers were "wrong," six were just in the wrong place. (Error first noted by Rodriguez Torres.)
Distribution of Carmichael Numbers (pp. 314-317)
316Error: End of paragraph two: "by Ketter in 1988" should be "by Keller in 1988." (Error noted by Dubner.)
Chapter 5: Which Special Kinds of Primes Have Been Considered?
Sophie Germain Primes (pp. 329-333)
331Error: Dubner's palindromic Sophie Germain primes should be
p = 39493939493 and 2p+1 = 78987878987. (Error noted by Rodriguez Torres, corrected by Dubner).
Addendum on Cullen Numbers (pp. 360-361)
360Error: Cn is prime for n=6611 (not 5611). (Error noted by Keller.)
360Error: (Third paragraph) "Dubner and Riesel studied divisibility properties..." should be "Dubner and Keller studied..." (Error noted by Dubner.)
361Error: C'n is prime for n=822 (not 882). (Error noted by Keller.)
361Addition: Keller's 1995 paper referred to has appeared:
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 = 10n-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*(23567 - 21189) - 6*21189, Tony Forbes, 1997.
510 Addition: Prime 6-tuplets, with 155 digits, p, p+4, p+6, p+10, p+12, p+16, with p = 2512 + 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 22.1038+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.
