On the number of prime factors of integers of the form ab + 1

K. Győry; A. Sárközy; C. L. Stewart

Acta Arithmetica (1996)

  • Volume: 74, Issue: 4, page 365-385
  • ISSN: 0065-1036

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K. Győry, A. Sárközy, and C. L. Stewart. "On the number of prime factors of integers of the form ab + 1." Acta Arithmetica 74.4 (1996): 365-385. <http://eudml.org/doc/206859>.

@article{K1996,
author = {K. Győry, A. Sárközy, C. L. Stewart},
journal = {Acta Arithmetica},
keywords = {upper bounds for the greatest prime factor; large sieve inequality; Siegel-Walfisz theorem; decomposable form equations; number of distinct prime factors of an integer},
language = {eng},
number = {4},
pages = {365-385},
title = {On the number of prime factors of integers of the form ab + 1},
url = {http://eudml.org/doc/206859},
volume = {74},
year = {1996},
}

TY - JOUR
AU - K. Győry
AU - A. Sárközy
AU - C. L. Stewart
TI - On the number of prime factors of integers of the form ab + 1
JO - Acta Arithmetica
PY - 1996
VL - 74
IS - 4
SP - 365
EP - 385
LA - eng
KW - upper bounds for the greatest prime factor; large sieve inequality; Siegel-Walfisz theorem; decomposable form equations; number of distinct prime factors of an integer
UR - http://eudml.org/doc/206859
ER -

References

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  1. [1] E. R. Canfield, P. Erdős and C. Pomerance, On a problem of Oppenheim concerning 'Factorisatio Numerorum', J. Number Theory 17 (1983), 1-28. Zbl0513.10043
  2. [2] H. Davenport, Multiplicative Number Theory, 2nd ed., Graduate Texts in Math. 74, Springer, 1980. Zbl0453.10002
  3. [3] P. Erdős, C. Pomerance, A. Sárközy and C. L. Stewart, On elements of sumsets with many prime factors, J. Number Theory 44 (1993), 93-104. Zbl0780.11040
  4. [4] P. Erdős, C. L. Stewart and R. Tijdeman, Some diophantine equations with many solutions, Compositio Math. 66 (1988), 37-56. Zbl0639.10014
  5. [5] P. Erdős and P. Turán, On a problem in the elementary theory of numbers, Amer. Math. Monthly 41 (1934), 608-611. Zbl60.0917.05
  6. [6] J.-H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561-584. Zbl0521.10015
  7. [7] J.-H. Evertse, The number of solutions of decomposable form equations, to appear. Zbl0886.11015
  8. [8] J.-H. Evertse and K. Győry, Finiteness criteria for decomposable form equations, Acta Arith. 50 (1988), 357-379. Zbl0595.10013
  9. [9] P. X. Gallagher, The large sieve, Mathematika 14 (1967), 14-20. Zbl0163.04401
  10. [10] K. Győry, On the numbers of families of solutions of systems of decomposable form equations, Publ. Math. Debrecen 42 (1993), 65-101. Zbl0792.11004
  11. [11] K. Győry, Some applications of decomposable form equations to resultant equations, Colloq. Math. 65 (1993), 267-275. Zbl0820.11018
  12. [12] K. Győry, C. L. Stewart and R. Tijdeman, On prime factors of sums of integers I, Compositio Math. 59 (1986), 81-88. Zbl0602.10031
  13. [13] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, 1979. Zbl0423.10001
  14. [14] C. Pomerance, A. Sárközy and C. L. Stewart, On divisors of sums of integers III, Pacific J. Math. 133 (1988), 363-379. Zbl0668.10055
  15. [15] A. Sárközy, Hybrid problems in number theory, in: Number Theory, New York 1985-88, Lecture Notes in Math. 1383, Springer, 1989, 146-169. 
  16. [16] A. Sárközy, On sums a + b and numbers of the form ab + 1 with many prime factors, Grazer Math. Ber. 318 (1992), 141-154. 
  17. [17] A. Sárközy and C. L. Stewart, On divisors of sums of integers V, Pacific J. Math. 166 (1994), 373-384. Zbl0841.11049
  18. [18] A. Sárközy and C. L. Stewart, On prime factors of integers of the form ab + 1, to appear. Zbl0960.11045
  19. [19] H. P. Schlickewei, S-unit equations over number fields, Invent. Math. 102 (1990), 95-107. Zbl0711.11017
  20. [20] H. P. Schlickewei, The quantitative Subspace Theorem for number fields, Compositio Math. 82 (1992), 245-273. Zbl0751.11033
  21. [21] W. M. Schmidt, The subspace theorem in diophantine approximations, Compositio Math. 69 (1989), 121-173. Zbl0683.10027
  22. [22] C. L. Stewart, Some remarks on prime divisors of sums of integers, in: Séminaire de Théorie des Nombres, Paris, 1984-85, Progr. Math. 63, Birkhäuser, 1986, 217-223. 
  23. [23] C. L. Stewart and R. Tijdeman, On prime factors of sums of integers II, in: Diophantine Analysis, J. H. Loxton and A. J. van der Poorten (eds.), Cambridge University Press, 1986, 83-98. Zbl0602.10032

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