On the Diophantine equation q n - 1 q - 1 = y

Amir Khosravi; Behrooz Khosravi

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 1, page 1-7
  • ISSN: 0010-2628

Abstract

top
There exist many results about the Diophantine equation ( q n - 1 ) / ( q - 1 ) = y m , where m ≥ 2 and n ≥ 3 . In this paper, we suppose that m = 1 , n is an odd integer and q a power of a prime number. Also let y be an integer such that the number of prime divisors of y - 1 is less than or equal to 3 . Then we solve completely the Diophantine equation ( q n - 1 ) / ( q - 1 ) = y for infinitely many values of y . This result finds frequent applications in the theory of finite groups.

How to cite

top

Khosravi, Amir, and Khosravi, Behrooz. "On the Diophantine equation $\frac{q^n-1}{q-1}=y$." Commentationes Mathematicae Universitatis Carolinae 44.1 (2003): 1-7. <http://eudml.org/doc/249149>.

@article{Khosravi2003,
abstract = {There exist many results about the Diophantine equation $(q^n-1)/(q-1)=y^m$, where $m\ge 2$ and $n\ge 3$. In this paper, we suppose that $m=1$, $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y-1$ is less than or equal to $3$. Then we solve completely the Diophantine equation $(q^n-1)/(q-1)=y$ for infinitely many values of $y$. This result finds frequent applications in the theory of finite groups.},
author = {Khosravi, Amir, Khosravi, Behrooz},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {higher order Diophantine equation; exponential Diophantine equation; exponential Diophantine equation; Fermat prime; Mersenne prime},
language = {eng},
number = {1},
pages = {1-7},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the Diophantine equation $\frac\{q^n-1\}\{q-1\}=y$},
url = {http://eudml.org/doc/249149},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Khosravi, Amir
AU - Khosravi, Behrooz
TI - On the Diophantine equation $\frac{q^n-1}{q-1}=y$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 1
SP - 1
EP - 7
AB - There exist many results about the Diophantine equation $(q^n-1)/(q-1)=y^m$, where $m\ge 2$ and $n\ge 3$. In this paper, we suppose that $m=1$, $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y-1$ is less than or equal to $3$. Then we solve completely the Diophantine equation $(q^n-1)/(q-1)=y$ for infinitely many values of $y$. This result finds frequent applications in the theory of finite groups.
LA - eng
KW - higher order Diophantine equation; exponential Diophantine equation; exponential Diophantine equation; Fermat prime; Mersenne prime
UR - http://eudml.org/doc/249149
ER -

References

top
  1. Bennett M., Rational approximation to algebraic number of small height: The Diophantine equation | a x n - b y n | = 1 , J. Reine Angew Math. 535 (2001), 1-49. (2001) Zbl1009.05033MR1837094
  2. Bugeaud Y., Linear forms in p -adic logarithms and the Diophantine equation ( x n - 1 ) / ( x - 1 ) = y q , Math. Proc. Cambridge Philos. Soc. 127 (1999), 373-381. (1999) MR1713116
  3. Bugeaud Y., Laurent M., Minoration effective de la distance p -adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), 311-342. (1996) Zbl0870.11045MR1423057
  4. Bugeaud Y., Mignotte M., On integers with identical digits, Mathematika 46 (1999), 411-417. (1999) Zbl1033.11012MR1832631
  5. Bugeaud Y., Mignotte M., Roy Y., Shorey T.N., On the Diophantine equation ( x n - 1 ) / ( x - 1 ) = y q , Math. Proc. Cambridge Philos. Soc. 127 (1999), 353-372. (1999) MR1713115
  6. Bugeaud Y., Mignotte M., Roy Y., On the Diophantine equation ( x n - 1 ) / ( x - 1 ) = y q , Pacific J. Math. 193 (2) (2000), 257-268. (2000) MR1755817
  7. Bugeaud Y., Hanrot G., Mignotte M., Sur l’equation diophantiene ( x n - 1 ) / ( x - 1 ) = y q III, (French), Proc. London Math. Soc. III. Ser. 84 (1) (2002), 59-78. (2002) MR1863395
  8. Crescenzo P., A Diophantine equation arises in the theory of finite groups, Adv. Math. 17 (1975), 25-29. (1975) MR0371812
  9. Edgar H., Problems and some results concerning the Diophantine equation 1 + A + A 2 + ⋯ + A x - 1 = P y , Rocky Mountain J. Math. 15 (1985), 327-329. (1985) MR0823244
  10. Guralnick R.M., Subgroups of prime power index in a simple group, J. Algebra 81 (1983), 304-311. (1983) Zbl0515.20011MR0700286
  11. Hardy G.H., Wright E.M., An Introduction to Theory of Numbers, Oxford University Press, 1962. 
  12. Le M., A note on the Diophantine equation ( x m - 1 ) / ( x - 1 ) = y n , Acta Arith. 64 (1993), 19-28. (1993) Zbl0802.11011MR1220482
  13. Le M., A note on perfect powers of the form x m - 1 + ⋯ + x + 1 , Acta Arith. 69 (1995), 91-98. (1995) Zbl0819.11012MR1310844
  14. Ljunggren W., Noen setninger om ubestemte likninger av formen ( x n - 1 ) / ( x - 1 ) = y q , Norsk. Mat. Tidsskr. 25 (1943), 17-20. (1943) 
  15. Mollin R.A., Fundamental Number Theory with Applications, CRC Press, New York, 1998. MR2404578
  16. Nagell T., Note sur l’equation indéterminée ( x n - 1 ) / ( x - 1 ) = y q , Norsk. Mat. Tidsskr. 2 (1920), 75-78. (1920) 
  17. Saradha N., Shorey T.N., The equation ( x n - 1 ) / ( x - 1 ) = y q with x square,, Math. Proc. Cambridge Philos. Soc. 125 (1999), 1-19. (1999) MR1645497
  18. Shorey T.N., Exponential Diophantine equation involving product of consecutive integers and related equations, (English) Bambah, R.P. (Ed.) et al., Number theory; Basel, Birkhäuser, Trends in Mathematics, (2000), 463-495. MR1764814
  19. Shorey T.N., Tijdeman R., New applications of Diophantine approximation to Diophantine equations, Math. Scand. 39 (1976), 5-18. (1976) MR0447110
  20. Shorey T.N., Exponential Diophantine equations, Cambridge Tracts in Mathematics 87 (1986), Cambridge University Press, Cambridge. Zbl1156.11015MR0891406
  21. Yu L., Le M., On the Diophantine equation ( x m - 1 ) / ( x - 1 ) = y n , Acta Arith. 83 (1995), 363-366. (1995) Zbl0870.11019

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.