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Keywords:
higher order Diophantine equation; exponential Diophantine equation
higher order Diophantine equation; exponential Diophantine equation
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.
References:
[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. MR 1837094 | Zbl 1009.05033
[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
[3] Bugeaud Y., Laurent M.: Minoration effective de la distance $p$-adique entre puissances de nombres algébriques. J. Number Theory 61 (1996), 311-342. MR 1423057 | Zbl 0870.11045
[4] Bugeaud Y., Mignotte M.: On integers with identical digits. Mathematika 46 (1999), 411-417. MR 1832631 | Zbl 1033.11012
[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
[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
[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
[8] Crescenzo P.: A Diophantine equation arises in the theory of finite groups. Adv. Math. 17 (1975), 25-29. MR 0371812
[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
[10] Guralnick R.M.: Subgroups of prime power index in a simple group. J. Algebra 81 (1983), 304-311. MR 0700286 | Zbl 0515.20011
[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. MR 1220482 | Zbl 0802.11011
[13] Le M.: A note on perfect powers of the form $x^{m-1}+\cdots+x+1$. Acta Arith. 69 (1995), 91-98. MR 1310844 | Zbl 0819.11012
[14] Ljunggren W.: Noen setninger om ubestemte likninger av formen $(x^n-1)/(x-1)=y^q$. Norsk. Mat. Tidsskr. 25 (1943), 17-20.
[15] Mollin R.A.: Fundamental Number Theory with Applications. CRC Press, New York, 1998. MR 2404578
[16] Nagell T.: Note sur l'equation indéterminée $(x^n-1)/(x-1)=y^q$. Norsk. Mat. Tidsskr. 2 (1920), 75-78.
[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
[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
[19] Shorey T.N., Tijdeman R.: New applications of Diophantine approximation to Diophantine equations. Math. Scand. 39 (1976), 5-18. MR 0447110
[20] Shorey T.N.: Exponential Diophantine equations. Cambridge Tracts in Mathematics 87 (1986), Cambridge University Press, Cambridge. MR 0891406 | Zbl 1156.11015
[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
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