Borevich, Z. I.; Shafarevich, I. R. Number theory. (Teoriya chisel). 3rd compl. ed. (Russian) Zbl 0592.12001 Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 504 p. R. 4.00 (1985). In this third, complemented edition there are many changes throughout the text reflecting new important results closely adjoined to formerly discussed subjects. [For a review of the first edition see Zbl 0121.042.] The reviewer’s remarks: 1) The authors’ account of the results of L. M. Adleman and D. R. Heath-Brown [Invent. Math. 79, 409-416 (1985; Zbl 0557.10034)] and of E. Fouvry [ibid. 79, 383-407 (1985; Zbl 0557.10035)] on Fermat’s last theorem (see p. 279) is incorrect. 2) In connection with a Hecke conjecture (see p. 402) the authors do not cite the paper of T. Shintani [J. Fac. Sci., Univ. Tokyo, Sect. I A 23, 393-417 (1976; Zbl 0349.12007)] who has proved this conjecture. Reviewer: O.M.Fomenko Cited in 4 ReviewsCited in 53 Documents MSC: 12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory 11Axx Elementary number theory 11Dxx Diophantine equations 11Rxx Algebraic number theory: global fields 11Sxx Algebraic number theory: local fields Keywords:congruence; quadratic field; algebraic complement; indefinite; equation; factorizable form; Fermat’s last theorem; Hecke conjecture Citations:Zbl 0121.042; Zbl 0557.10034; Zbl 0557.10035; Zbl 0349.12007 × Cite Format Result Cite Review PDF Online Encyclopedia of Integer Sequences: Primes of form 8n+1, that is, primes congruent to 1 mod 8. Primes congruent to {0, 1, 4} mod 5. Odd primes p such that 13 is a square mod p. Primes p such that 17 is a square mod p. Primes of the form x^2 + 45y^2. Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative. Duplicate of A139492. Primes of the form 4*x^2+6*x*y-7*y^2. Primes of the form x^2+12*x*y-y^2. Primes of the form 9*x^2+7*x*y-5*y^2. Primes of the form x^2+15*x*y-y^2. Primes of the form 8*x^2+x*y-8*y^2. Primes of the form 4*x^2+9*x*y-11*y^2. Primes of the form x^2+4*x*y-2*y^2 (as well as of the form 3*x^2+6*x*y+y^2). Primes of the form -x^2+4*x*y+2*y^2 (as well as of the form 5*x^2+8*x*y+2*y^2). Primes of the form 2*x^2+2*x*y-3*y^2 (as well as of the form 2*x^2+6*x*y+y^2). Primes of the form -2*x^2+2*x*y+3*y^2 (as well as of the form 6*x^2+10*x*y+3*y^2). Primes of the form 2*x^2 + 3*x*y - 3*y^2 (as well as of the form 6*x^2 + 9*x*y + 2*y^2). Primes of the form -2*x^2 + 3*x*y + 3*y^2 (as well as of the form 4*x^2 + 7*x*y + y^2). Primes of the form 3*x^2 + 2*x*y - 3*y^2 (as well as of the form 3*x^2 + 8*x*y + 2*y^2). Primes of the form x^2+6*x*y-y^2 (as well as of the form 6*x^2+8*x*y+y^2). Primes of the form x^2+6*x*y-2*y^2 (as well as of the form 5*x^2+8*x*y+y^2). Primes of the form 2*x^2+4*x*y-5*y^2 (as well as of the form 2*x^2+8*x*y+y^2). Primes of the form 3*x^2+3*x*y-4*y^2 (as well as of the form 8*x^2+11*x*y+2*y^2). Primes of the form -3*x^2+3*x*y+4*y^2 (as well as of the form 6*x^2+9*x*y+y^2). Primes of the form 3*x^2+5*x*y-3*y^2 (as well as 5*x^2+9*x*y+y^2). Primes of the form 2*x^2+6*x*y-3*y^2 (as well as of the form 5*x^2+10*x*y+2*y^2). Primes of the form -2*x^2+6*x*y+3*y^2 (as well as of the form 7*x^2+12*x*y+3*y^2). Primes of the form x^2+9*x*y-3*y^2 (as well as of the form 7*x^2+11*x*y+y^2). Primes of the form -x^2+9*x*y+3*y^2 (as well as of the form 11*x^2+15*x*y+3*y^2). Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2). Primes of the form -x^2 + 8*x*y + 8*y^2 (as well as of the form 15*x^2 + 24*x*y + 8*y^2). Primes of the form 4*x^2 + 3*x*y - 4*y^2 (as well as of the form 2*x^2 + 9*x*y + y^2). Primes of the form 3*x^2 + 5*x*y - 5*y^2 (as well as of the form 7*x^2 + 13*x*y + 3*y^2). Primes of the form x^2 + 9*x*y - y^2 (as well as of the form 9*x^2 + 11*x*y + y^2). Primes of the form 3*x^2 + 4*x*y - 6*y^2 (as well as of the form 3*x^2 + 10*x*y + y^2). Primes of the form -3*x^2 + 4*x*y + 6*y^2 (as well as of the form 7*x^2 + 12*x*y + 2*y^2). Primes of the form 4*x^2 + 3*x*y - 5*y^2 (as well as of the form 8*x^2 + 11*x*y + y^2). Primes of the form -x^2 + 5*x*y + 5*y^2 (as well as of the form 9*x^2 + 15*x*y + 5*y^2). Primes of the form 3*x^2 + 4*x*y - 5*y^2 (as well as of the form 3*x^2 + 10*x*y + 2*y^2). Duplicate of A038987.