Ribenboim, Paulo The new book of prime number records. 3rd ed. (English) Zbl 0856.11001 New York, NY: Springer-Verlag. xxiv, 541 p. (1996). This monograph is the updated and enlarged 3rd edition of a well-known, informative, entertaining book, which can be read by number theorists with pleasure. The first edition (1988) was reviewed by H. C. Williams in Zbl 0642.10001. Due to the fast development in computational number theory, records given in the first edition are often out of date, and so, as the author writes in his elegant, witty style: “The new book of prime number records differs little from its predecessor in the general planning. But it contains new sections and updated records. It has been comforting to learn about the countless computers (machines and men), grinding without stop, so that more lines with new large numbers could be added, bringing despair for the printers and proofreaders.” In more detail, this monograph deals with elementary proofs for the infiniteness of the primes, with a 150-page-study of primality tests, with prime-representing functions (including “prime-producing” polynomials), with special primes (regular primes, Wieferich primes, “Sophie Germain primes”, all connected with “Fermat’s last theorem”; unfortunately E. Fouvry’s deep result on the first case of Fermat’s last theorem (“Sophie Germain primes”) is not mentioned), and with heuristic and probabilistic results on prime numbers. The extensive bibliography covers more than 70 pages. Of course, it is not possible to describe in a short review the changes in the text improving the second edition (Example: the proof of \(\pi (x)\to \infty\), using Fermat’s numbers \(F_n= 2^{2^n}+ 1\), formerly attributed to Pólya, is, as the author points out, in fact due to Goldbach). The addenda make up for an enlargement of the size of the monograph by more than 10%. It ought to be mentioned that the printing of the 3rd edition is much nicer than that of the foregoing edition. Hopefully this monograph will induce many students to become interested in prime number theory. Reviewer: Wolfgang Schwarz (Frankfurt am Main) Cited in 3 ReviewsCited in 111 Documents MSC: 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11A41 Primes 11N05 Distribution of primes 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11N13 Primes in congruence classes 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11P32 Goldbach-type theorems; other additive questions involving primes 11Y11 Primality Keywords:prime number theorem; large primes; twin primes; Goldbach conjecture; primality tests; elementary prime number theory; prime representing functions; distribution of the primes; additive prime number theory; special primes; heuristic arguments concerning primes; extensive bibliography Citations:Zbl 0842.11001; Zbl 0642.10001 × Cite Format Result Cite Review PDF Online Encyclopedia of Integer Sequences: Euler totient function phi(n): count numbers <= n and prime to n. The prime numbers. Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. Primorial plus 1 primes: primes p such that 1 + product of primes up to p is prime. Smallest m >= 0 such that (2n-1)2^m-1 is prime, or -1 if no such value exists. Smallest positive quadratic nonresidue modulo p, where p is the n-th prime. Least positive integer k for which the Jacobi symbol (k|2*n-1) is less than 1, where 2*n-1 is a nonsquare; a(n)=0 if 2*n-1 is a square. Start of the first run of exactly n consecutive primes, none of which are twin primes. Number of primes in the interval [p(n), p(n)^2] minus p(n), where p(n) is the n-th prime. Write the numbers from 1 to n^2 consecutively in n rows of length n; a(n) = minimal number of primes in a row. Number of primes between n^2 and (n+1/2)^2. Number of primes between n^2-n and n^2 (inclusive). Smallest odd pseudoprime k > b to bases p_i, i.e., the smallest composite number k > b such that p_i^(k-1)-1 is divisible by k, p_i are the prime factors of b, where b is the n-th squarefree number, A005117(n). Numbers k such that 1 + (k-1)^2 and ((k-1)/2)^2 + ((k+1)/2)^2 = (1/2)*(k^2+1) are primes. Numbers which yield primes when ”13” is prefixed or appended: N natural number is a member of the sequence, if P=”13N” (prefixed 13) and A=”N13” (appended 13) are prime. Primes whose squares are a concatenation of 2 with some prime. Number of primes q with (2m)^2+1 <= q < (2m+1)^2-2m. Number of primes p with (2m+1)^2 - 2m <= p < (2m+1)^2. Number of primes q with (2n-1)^2+1 <= q < (2n)^2-(2n-1). Smallest prime concatenation of the first n primes, or 0 if no such prime exists. Largest prime concatenation of the first n primes, or 0 if no such prime exists. Primes which are the concatenation of two squares (in decimal notation). Naturals n for which 1 + 10*n^3 (A168147) is prime. ”1-ply” palindromic primes; see Comments. ”2-ply” palindromic primes. ”3-ply” palindromic primes. Numbers k such that k+j is prime for every j, where 1 <= j < k and gcd(j,k) = 1.