Some diophantine equations with many solutions

 Access to full text

 Full (PDF)

1. 1 A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms, Philos. Trans. Roy. Soc. London Ser.A263 (1967/68) 173-191. Zbl0157.09702MR228424
2. 2 A. Balog and A. Sárközy, On sums of sequences of integers, II, Acta Math. Hungar.44 (1984) 169-179. Zbl0559.10034MR759044
3. 3 E. Bombieri and W.M. Schmidt,On Thue's equation, Invent. Math.88 (1987) 69-81. Zbl0614.10018MR877007
4. 4 N.G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.) 15 (1951) 25-32. Zbl0043.06502MR43838
5. 5 N.G. de Bruijn, On the number of positive integers ≤ x and free of prime factors > y, Nederl. Akad. Wetensch. Proc. Ser.A54 (1951) 50-60. Zbl0042.04204
6. 6 A.A. Buchstab, On those numbers in an arithmetical progression all prime factors of which are small in order of magnitude (Russian), Dokl. Akad. Nauk. SSSR67 (1949) 5-8. MR30995
7. 7 E.R. Canfield, P. Erdös and C. Pomerance, On a problem of Oppenheim concerning 'Factorisatio Numerorum', J. Number Th.17 (1983) 1-28. Zbl0513.10043MR712964
8. 8 J.-M.-F. Chamayou, A probabilistic approach to a differential-difference equation arising in analytic number theory, Math. Comp.27 (1973) 197-203. Zbl0252.65066MR336952
9. 9 J. Coates, An effective p-adic analogue of a theorem of Thue, Acta Arith.15 (1968/69) 279-305. Zbl0221.10025MR242768
10. 10 J. Coates, An effective p-adic analogue of a theorem of Thue II, The greatest prime factor of a binary form, Acta Arith.16 (1969/70) 399-412. Zbl0221.10026MR263741
11. 11 H. Davenport and K.F. Roth, Rational approximations to algebraic numbers, Mathematika2 (1955) 160-167. Zbl0066.29302MR77577
12. 12 K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys.22 (1930) A10, 1-14. Zbl56.0178.04JFM56.0178.04
13. 13 P. Erdös, The difference of consecutive primes, Duke Math. J.6 (1940) 438-441. Zbl0023.29801MR1759JFM66.0162.04
14. 14 P. Erdös, Problems in number theory and combinatorics, Proc. 6th Manitoba Conference on Numerical Mathematics, Congress Numer. 18, Utilitas Math., Winnipeg, Man. (1977) 35-58. Zbl0471.10002MR532690
15. 15 P. Erdös and P. Turán, On a problem in the elementary theory of numbers, Amer. Math. Monthly41 (1934) 608-611. Zbl0010.29401MR1523239JFM60.0917.05
16. 16 J.-H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math.75 (1984) 561-584. Zbl0521.10015MR735341
17. 17 J.-H. Evertse and K. Györy, On unit equations and decomposable form equations, J. reine angew. Math.358 (1985) 6-19. Zbl0552.10010MR797671
18. 18 J.-H. Evertse, K. Györy, C.L. Stewart and R. Tijdeman, S-unit equations and their applications, New Advances in Transcendence Theory (to appear). Zbl0658.10023MR971998
19. 19 J.-H. Evertse, K. Györy, C.L. Stewart and R. Tijdeman, On S-unit equations in two unknowns, Invent. Math. (to appear). Zbl0662.10012MR939471
20. 20 K. Györy, Explicit upper bounds for the solutions of some Diophantine equations, Ann. Acad. Sc. Fenn. Ser. AI5 (1980) 3-12. Zbl0402.10018MR595172
21. 21 K. Györy and Z.Z. Papp, Effective estimates for the integer solutions of norm form and discriminant form equations, Publ. Math. Debrecen25 (1978) 311-325. Zbl0394.10010MR517016
22. 22 K. Györy and Z.Z. Papp, Norm form equations and explicit lower bounds for linear forms with algebraic coefficients, Studies in Pure Math. to the Memory of Paul Turán, Birkhäuser, Basel, pp. 245-257. Zbl0518.10020MR820227
23. 23 K. Györy, C.L. Stewart and R. Tijdeman, On prime factors of sums of integers I, Comp. Math.59 (1986) 81-88. Zbl0602.10031MR850123
24. 24 K. Györy, C.L. Stewart and R. Tijdeman, On prime factors of sums of integers III, Acta Arith. (to appear). Zbl0588.10001MR932530
25. 25 G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th edn., Oxford (1979). Zbl0020.29201MR568909
26. 26 A.E. Ingham, On the difference between consecutive primes, Quarterly J. Math. Oxford8 (1937) 255-266. Zbl0017.38904JFM63.0903.04
27. 27 S. Lang, Integral points on curves, Publ. Math. I.H.E.S.6 (1960) 27-43. Zbl0112.13402MR130219
28. 28 D.J. Lewis and K. Mahler, On the representations of integers by binary forms, Acta Arith.6 (1960) 333-363. Zbl0102.03601MR120195
29. 29 J van de Lune and E. Wattel, On the numerical solution of a differential - difference equation arising in analytic number theory, Math. Comp.23 (1969) 417-421. Zbl0176.46602MR247789
30. 30 K. Mahler, Zur Approximation algebraischer Zahlen, I: Über den grössten Primteiler binärer Formen, Math. Ann.107 (1933) 691-730. Zbl0006.10502MR1512822JFM59.0220.01
31. 31 K. Mahler, Zur Approximation algebraischer Zahlen, II: Über die Anzahl der Darstellungen grösser Zahlen durch binäre Formen, Math. Ann.108 (1933) 37-55. Zbl0006.15604MR1512833JFM59.0220.01
32. 32 K. Mahler, On the lattice points on curves of genus 1, Proc. London Math. Soc. (2) 39 (1935) 431-466. Zbl0012.15006JFM61.0146.02
33. 33 K. Mahler, On Thue's equation, Math. Scand.55 (1984) 188-200. Zbl0544.10014
34. 34 H. Maier, Chains of large gaps between consecutive primes, Adv. in Math.39 (1981) 257-269. Zbl0457.10023MR614163
35. 35 K.K. Norton, Numbers with small prime factors, and the least k-th power non-residue, Memoirs of the American Math. Soc.106 (1971). Zbl0211.37801MR286739
36. 36 A. Sárközy and C.L. Stewart, On divisors of sums of integers I, Acta Math. Hungar.48 (1986) 147-154. Zbl0612.10042MR858392
37. 37 A. Sárközy and C.L. Stewart, On divisors of sums of integers II, J. reine angew. Math.365 (1986) 171-191. Zbl0578.10045MR826157
38. 38 J.H. Silverman, Integer points and the rank of Thue elliptic curves, Invent. Math.66 (1982) 395-404. Zbl0494.14008MR662599
39. 39 J.H. Silverman, Representations of integers by binary forms and the rank of the Mordell-Weil group, Invent. Math.74 (1983) 281-292. Zbl0525.14012MR723218
40. 40 J.H. Silverman, Integer points on curves of genus 1, J. London Math. Soc.28 (1983) 1-7. Zbl0487.10015MR703458
41. 41 J.H. Silverman, Quantitative results in Diophantine geometry, Preprint, M.I.T. (1984). 
42. 42 V.G. Sprindzhuk, Estimation of the solutions of the Thue equation (Russian), Izv. Akad. Nauk. SSSR Ser. Mat.36 (1972) 712-741. Zbl0244.10030MR313186
43. 43 C.L. Stewart,, Some remarks on prime divisors of integers, Seminaire de Théorie des Nombres, Paris1984-85, Progress in Math.63, Birkhauser, Boston, etc. (1986) 217-223. Zbl0602.10030MR897351
44. 44 C.L. Stewart and R. Tijdeman, On prime factors of sums of integers II, Diophantine Analysis, LMS Lecture Notes109, Cambridge Univ. Press (1986) 83-98. Zbl0602.10032MR874122

1. K. Győry, A. Sárközy, On prime factors of integers of the form (ab+1)(bc+1)(ca+1)
2. K. Győry, A. Sárközy, C. L. Stewart, On the number of prime factors of integers of the form ab + 1
3. Jeffrey C. Lagarias, Kannan Soundararajan, Smooth solutions to the $abc$ equation: the $xyz$ Conjecture