The diophantine equation x² + C = yⁿ

J. H. E. Cohn

Acta Arithmetica (1993)

  • Volume: 65, Issue: 4, page 367-381
  • ISSN: 0065-1036

How to cite

top

J. H. E. Cohn. "The diophantine equation x² + C = yⁿ." Acta Arithmetica 65.4 (1993): 367-381. <http://eudml.org/doc/206586>.

@article{J1993,
author = {J. H. E. Cohn},
journal = {Acta Arithmetica},
keywords = {exponential diophantine equation},
language = {eng},
number = {4},
pages = {367-381},
title = {The diophantine equation x² + C = yⁿ},
url = {http://eudml.org/doc/206586},
volume = {65},
year = {1993},
}

TY - JOUR
AU - J. H. E. Cohn
TI - The diophantine equation x² + C = yⁿ
JO - Acta Arithmetica
PY - 1993
VL - 65
IS - 4
SP - 367
EP - 381
LA - eng
KW - exponential diophantine equation
UR - http://eudml.org/doc/206586
ER -

References

top
  1. [1] A. Aigner, Die diophantische Gleichung x ² + 4 D = y p im Zusammenhang mit Klassenzahlen, Monatsh. Math. 72 (1968), 1-5. Zbl0164.04805
  2. [2] J. Blass, A note on diophantine equation Y²+k=X⁵, Math. Comp. 30 (1976), 638-640. Zbl0336.10012
  3. [3] J. Blass and R. Steiner, On the equation y²+k=x⁷, Utilitas Math. 13 (1978), 293-297. Zbl0375.10010
  4. [4] E. Brown, Diophantine equations of the form x²+D=yⁿ, J. Reine Angew. Math. 274/275 (1975), 385-389. Zbl0303.10014
  5. [5] E. Brown, Diophantine equations of the form a x ² + D b ² = y p , J. Reine Angew. Math. 291 (1977), 118-127. Zbl0338.10018
  6. [6] K. Chao, On the diophantine equation x²=yⁿ+1, xy≠0, Sci. Sinica (Notes) 14 (1964), 457-460. 
  7. [7] F. B. Coghlan and N. M. Stephens, The diophantine equation x³-y²=k, in: Computers in Number Theory, Academic Press, London, 1971, 199-205. 
  8. [8] E. L. Cohen, Sur l’équation diophantienne x ² + 11 = 3 k , C. R. Acad. Sci. Paris Sér. A 275 (1972), 5-7. Zbl0236.10008
  9. [9] E. L. Cohen, On the Ramanujan-Nagell equation and its generalizations, in: Proc. First Conference of the Canadian Number Theory Association, Banff, Alberta, 1988, de Gruyter, 1990, 81-92. 
  10. [10] J. H. E. Cohn, The Diophantine equation x²+3=yⁿ, Glasgow Math. J. 35 (1993), 203-206. 
  11. [11] J. H. E. Cohn, The diophantine equation x²+19=yⁿ, Acta Arith. 61 (1992), 193-197. Zbl0770.11018
  12. [12] J. H. E. Cohn, The diophantine equation x²+2^k=yⁿ, Arch. Math. (Basel) 59 (1992), 341-344. Zbl0770.11019
  13. [13] J. H. E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc. 7 (1965), 24-28. Zbl0127.01902
  14. [14] L. Euler, Algebra, Vol. 2. 
  15. [15] O. Korhonen, On the Diophantine equation Ax²+8B=yⁿ, Acta Univ. Oulu. Ser. A Sci. Rerum Natur. Math. 16 (1979). Zbl0414.10007
  16. [16] O. Korhonen, On the Diophantine equation Ax²+2B=yⁿ, Acta Univ. Oulu. Ser. A Sci. Rerum Natur. Math. 17 (1979). Zbl0416.10012
  17. [17] O. Korhonen, On the Diophantine equation Cx²+D=yⁿ, Acta Univ. Oulu. Ser. A Sci. Rerum Natur. Math. 25 (1981). Zbl0452.10018
  18. [18] V. A. Lebesgue, Sur l’impossibilité en nombres entiers de l’équation x m = y ² + 1 , Nouvelles Annales des Mathématiques (1) 9 (1850), 178-181. 
  19. [19] W. Ljunggren, On the diophantine equation x²+p²=yⁿ, Norske Vid. Selsk. Forh. Trondheim 16 (8) (1943), 27-30. Zbl0060.09106
  20. [20] W. Ljunggren, Über einige Arcustangensgleichungen die auf interessante unbestimmte Gleichungen führen, Ark. Mat. Astr. Fys. 29A (1943), no. 13. Zbl0028.10904
  21. [21] W. Ljunggren, On the diophantine equation x²+D=yⁿ, Norske Vid. Selsk. Forh. Trondheim 17 (23) (1944), 93-96. Zbl0060.09107
  22. [22] W. Ljunggren, On a diophantine equation, Norske Vid. Selsk. Forh. Trondheim 18 (32) (1945), 125-128. Zbl0060.09108
  23. [23] W. Ljunggren, New theorems concerning the diophantine equation Cx²+D=yⁿ, Norske Vid. Selsk. Forh. Trondheim 29 (1) (1956), 1-4. 
  24. [24] W. Ljunggren, On the diophantine equation y²-k=x³, Acta Arith. 8 (1963), 451-463. Zbl0132.28404
  25. [25] W. Ljunggren, On the diophantine equation Cx²+D=yⁿ, Pacific J. Math. 14 (1964), 585-596. Zbl0131.28401
  26. [26] W. Ljunggren, On the diophantine equation Cx²+D=2yⁿ, Math. Scand. 18 (1966), 69-86. Zbl0223.10007
  27. [27] W. Ljunggren, New theorems concerning the diophantine equation x ² + D = 4 y q , Acta Arith. 21 (1972), 183-191. Zbl0216.04101
  28. [28] L. J. Mordell, Diophantine Equations, Academic Press, London, 1969. 
  29. [29] T. Nagell, Sur l'impossibilité de quelques équations à deux indéterminées, Norsk. Mat. Forensings Skrifter No. 13 (1923), 65-82. 
  30. [30] T. Nagell, Løsning til oppgave nr 2, 1943, s. 29, Norske Mat. Tidsskrift 30 (1948), 62-64. 
  31. [31] T. Nagell, Verallgemeinerung eines Fermatschen Satzes, Arch. Math. (Basel) 5 (1954), 153-159. Zbl0055.03608
  32. [32] T. Nagell, Contributions to the theory of a category of diophantine equations of the second degree with two unknowns, Nova Acta Regiae Soc. Sci. Upsaliensis (4) 16 (2) (1955). 
  33. [33] T. Nagell, On the Diophantine equation x²+8D=yⁿ, Ark. Mat. 3 (1954), 103-112. 
  34. [34] S. Ramanujan, Question 464, J. Indian Math. Soc. 5 (1913), 120. 
  35. [35] T. N. Shorey, A. J. van der Poorten, R. Tijdeman and A. Schinzel, Applications of the Gel'fond-Baker method to diophantine equations, in: Transcendence Theory: Advances and Applications, Academic Press, London, 1977, 59-77. Zbl0371.10015
  36. [36] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986. Zbl0606.10011
  37. [37] B. M. E. Wren, y²+D=x⁵, Eureka 36 (1973), 37-38 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.