On the Diophantine equation \({y^{p} = \frac{f(x)}{g(x)}}\)

  • S. Subburam
  • A. Togbé


We consider the Diophantine equation
$$y^{p} = \frac{f(x)}{g(x)},$$
where \({x \in \mathbb{Z}}\) and \({y \in \mathbb{Q}}\) are unknowns, f(x) and g(x) are non-zero integer polynomials in variable x and p is prime. We give bounds for x, when \({(x, y) \in \mathbb{Z} \times \mathbb{Q}}\) is a solution of the equation. This improves the results of some recent papers.

Key words and phrases

Diophantine equation monic polynomial 

Mathematics Subject Classification

11D41 11D45 


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  1. 1.
    Balasubramanian, R., Shorey, T.N.: On the equation \((x+1)\cdots (x+k) = (y+1)\) \( \cdots (y+mk)\). Indag. Math., N. S. 4, 257–267 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monograph Enseign. Math., 28, Université de Genève (Geneva, 1980).Google Scholar
  3. 3.
    Fujiwara, M.: Über die obere Schranke des absoluten Betrages der Wurzelneiner algebraischen Gleichung. Tôhoku Math. J. 10, 167–171 (1916)zbMATHGoogle Scholar
  4. 4.
    Inkeri, K.: On the Diophantine equation \(a(x^n - 1)/(x - 1) = y^m\). Acta Arith. 21, 299–311 (1972)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ljunggren, W.: Noen setninger om ubestemte likninger av formen \((x^n - 1)/(x - 1) = y^q\). Norsk. Mat. Tidskr. 25, 17–20 (1943)Google Scholar
  6. 6.
    Luca, F., Walsh, P.G.: On a Diophantine equation related to a conjecture of Erdős and Graham. Glas. Mat. III(42), 281–289 (2007)CrossRefGoogle Scholar
  7. 7.
    T. Nagell, Note sur l'equation indéterminée \((x^n - 1)/(x-1) = y^q\), Norsk. Mat. Tidskr., (1920), 75–78Google Scholar
  8. 8.
    Poulakis, D.: A simple method for solving the Diophantine equation \(y^{2} = x^{4} + ax^{3} + bx^{2} + cx + d\). Elem. Math. 54, 32–36 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sankaranarayanan, A., Saradha, N.: Estimates for the solutions of certain Diophantine equations by Runge's method. Int. J. Number Theory 4, 475–493 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Shorey, T.N.: On the equation \(z^q=(x^n-1)/(x - 1)\). Indag. Math. 48, 345–351 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shorey, T.N., Tijdeman, R.: New applications of Diophantine approximations to Diophantine equation. Math. Scand. 39, 5–18 (1976)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Srikanth, R., Subburam, S.: The superelliptic equation \(y^{p} = x^{kp} + a_{kp - 1}x^{kp - 1} + \cdots + a_{0}\). J. Algebra and Number Theory Academia 2, 331–385 (2012)Google Scholar
  13. 13.
    Srikanth, R., Subburam, S.: On the Diophantine equation \(y^p = f(x_1, x_2, \ldots, x_n)\). Funct. Approx. Comment. Math. 58, 37–42 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. Srikanth and S. Subburam, On the Diophantine equation \(y^2 = \prod _{i \le 8}(x + k_{i})\), Proc. Indian Acad. Sci. (Math. Sci.), 128 (2018), article ID 41Google Scholar
  15. 15.
    Subburam, S.: The Diophantine equation \((y-q_1) (y-q_2)\cdots (y-q_m) = f (x)\). Acta Math. Hungar. 146, 40–46 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Szalay, L.: Fast algorithm for solving superelliptic equations of certain types, Acta Acad. Paedagog. Agriensis Sect. Mat. (N. S.) 27, 19–24 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Szalay, L.: Superelliptic equations of the form \(y^{p} = x^{kp} + a_{kp - 1}x^{kp - 1} +\cdots +a_{0}\). Bull. Greek Math. Soc. 46, 23–33 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Szalay, L.: Algorithm to solve ternary Diophantine equations. Turkish J. Math. 37, 733–738 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Tengely, Sz., Varga, N.: On a generalization of a problem of Erdős and Graham. Publ. Math. Debrecen 84, 475–482 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tengely, Sz., Varga N.: Rational function variant of a problem of Erdős and. Graham, Glasnik Mat. 50, 65–76 (2015)CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of MathematicsKalasalingam Academy of Research and EducationVirudhunagar DistrictIndia
  2. 2.Department of Mathematics, Statistics, and Computer SciencePurdue University NorthwestWestvilleUSA

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