| Title:
|
On the Diophantine equation $\frac{q^n-1}{q-1}=y$ (English) |
| Author:
|
Khosravi, Amir |
| Author:
|
Khosravi, Behrooz |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
1-7 |
| . |
| Category:
|
math |
| . |
| Summary:
|
There exist many results about the Diophantine equation $(q^n-1)/(q-1)=y^m$, where $m\ge 2$ and $n\geq 3$. 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. (English) |
| Keyword:
|
higher order Diophantine equation |
| Keyword:
|
exponential Diophantine equation |
| MSC:
|
11D41 |
| MSC:
|
11D61 |
| idZBL:
|
Zbl 1097.11015 |
| idMR:
|
MR2045841 |
| . |
| Date available:
|
2009-01-08T19:26:54Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119363 |
| . |
| Reference:
|
[1] Bennett M.: Rational approximation to algebraic number of small height: The Diophantine equation $|ax^n-by^n|=1$.J. Reine Angew Math. 535 (2001), 1-49. Zbl 1009.05033, MR 1837094 |
| Reference:
|
[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. MR 1713116 |
| Reference:
|
[3] Bugeaud Y., Laurent M.: Minoration effective de la distance $p$-adique entre puissances de nombres algébriques.J. Number Theory 61 (1996), 311-342. Zbl 0870.11045, MR 1423057 |
| Reference:
|
[4] Bugeaud Y., Mignotte M.: On integers with identical digits.Mathematika 46 (1999), 411-417. Zbl 1033.11012, MR 1832631 |
| Reference:
|
[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. MR 1713115 |
| Reference:
|
[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. MR 1755817 |
| Reference:
|
[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. MR 1863395 |
| Reference:
|
[8] Crescenzo P.: A Diophantine equation arises in the theory of finite groups.Adv. Math. 17 (1975), 25-29. MR 0371812 |
| Reference:
|
[9] Edgar H.: Problems and some results concerning the Diophantine equation $1+A+A^2+\cdots+A^{x-1}=P^y$.Rocky Mountain J. Math. 15 (1985), 327-329. MR 0823244 |
| Reference:
|
[10] Guralnick R.M.: Subgroups of prime power index in a simple group.J. Algebra 81 (1983), 304-311. Zbl 0515.20011, MR 0700286 |
| Reference:
|
[11] Hardy G.H., Wright E.M.: An Introduction to Theory of Numbers.Oxford University Press, 1962. |
| Reference:
|
[12] Le M.: A note on the Diophantine equation $(x^m-1)/(x-1)=y^n$.Acta Arith. 64 (1993), 19-28. Zbl 0802.11011, MR 1220482 |
| Reference:
|
[13] Le M.: A note on perfect powers of the form $x^{m-1}+\cdots+x+1$.Acta Arith. 69 (1995), 91-98. Zbl 0819.11012, MR 1310844 |
| Reference:
|
[14] Ljunggren W.: Noen setninger om ubestemte likninger av formen $(x^n-1)/(x-1)=y^q$.Norsk. Mat. Tidsskr. 25 (1943), 17-20. |
| Reference:
|
[15] Mollin R.A.: Fundamental Number Theory with Applications.CRC Press, New York, 1998. MR 2404578 |
| Reference:
|
[16] Nagell T.: Note sur l'equation indéterminée $(x^n-1)/(x-1)=y^q$.Norsk. Mat. Tidsskr. 2 (1920), 75-78. |
| Reference:
|
[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. MR 1645497 |
| Reference:
|
[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. MR 1764814 |
| Reference:
|
[19] Shorey T.N., Tijdeman R.: New applications of Diophantine approximation to Diophantine equations.Math. Scand. 39 (1976), 5-18. MR 0447110 |
| Reference:
|
[20] Shorey T.N.: Exponential Diophantine equations.Cambridge Tracts in Mathematics 87 (1986), Cambridge University Press, Cambridge. Zbl 1156.11015, MR 0891406 |
| Reference:
|
[21] Yu L., Le M.: On the Diophantine equation $(x^m-1)/(x-1)=y^n$.Acta Arith. 83 (1995), 363-366. Zbl 0870.11019 |
| . |
