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The al-husayn equation x 4 + y 2 = z 2

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Abstract

We study the set of all natural solutions of the equation x 4 + y 2 = z 2, obtain general formulas describing all such solutions, and prove their equivalence.

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References

  1. A. O. Gel’fond, Solution of Equations in Integers, in Popular Lectures inMathematics (Nauka, Moscow, 1978), Vol. 8 [in Russian].

    Google Scholar 

  2. W. Sierpinski, Pythagorean Triangles (Uchpedgiz, Moscow, 1959) [in Russian].

    Google Scholar 

  3. A. Baker, Introduction to Number Theory (Vysheishaya Shkola, Minsk, 1995) [in Russian].

    Google Scholar 

  4. I. G. Bashmakova and E. I. Slavutin, History of Diophantine Analysis from Diophantus to Fermat (Nauka, Moscow, 1984) [in Russian].

    MATH  Google Scholar 

  5. Diophantus of Alexandria, Arithmetic and Book on Polygonal Numbers (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  6. P. Fermat, Studies in Number Theory and Diophantine Analysis (Nauka, Moscow, 1992) [in Russian].

    Google Scholar 

  7. L. E. Dickson, History of the Theory of Numbers, in Diophantine Analysis (Carnegie Institution of Washington, New York, 1934), Vol. 2.

    Google Scholar 

  8. L. J. Mordell, Diophantine Equations, in Pure Appl.Math. (Academic Press, New York, 1969), Vol. 30.

    Google Scholar 

  9. S. Sh. Kozhegel’dinov, Elements of the Theory of Diophantine Equations: Exercises and Problems (Prometei, Moscow, 1993) [in Russian].

    Google Scholar 

  10. S. Sh. Kozhegel’dinov, “On basic Heronian triangles,” Mat. Zametki 55(2), 72–79 (1994) [Math. Notes 55 (2), 151–156 (1994)].

    MathSciNet  Google Scholar 

  11. S. Sh. Kozhegel’dinov, “On the equivalence of general formulas for basic solutions of a Diophantine equation,” in III Republican Memorial Conference Dedicated to Professor T. I. Amanov (1923–1978) Collection of Papers, Part 2 (KarGU, Karaganda, 1998), pp. 129–135 [in Russian].

    Google Scholar 

  12. S. Sh. Kozhegel’dinov, Classical Diophantine Equations in Three and More Variables, In 5 volumes: (Novosibirsk, 2002), Vol. 1 (Almaaty, 2004), Vol. 2 (Almaaty, 2006), Vol. 3 (Semei, 2008), Vol. 4 (Semei, 2008), Vol. 5 [in Russian].

  13. S. Sh. Kozhegel’dinov, “On the solution of equations in natural numbers,” in Scientific Conference Dedicated to Professor T. I. Amanov on the Occasion of his 70th Anniversary, Abstracts of Papers (SGPI, Semipalatinsk, 1993), pp. 17–19 [in Russian].

    Google Scholar 

  14. S. Sh. Kozhegel’dinov, “On the use of arithmetical functions,” in II International Conference “Algebraic, Probabilistic, Geometric, Combinatorial and Functional Methods in Number Theory”, Abstracts of Papers (VGU, Voronezh, 1995), p. 85 [in Russian].

    Google Scholar 

  15. S. Sh. Kozhegel’dinov, “On a Diophantine equation,” in III International Conference “Problems of Current Interest in Number Theory and Its Applications”, Abstracts of Papers (TGPU, Tula, 1996), p. 77 [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Semipalatinsk State Pedagogical Institute, Semipalatinsk, Russia

    S. Sh. Kozhegel’dinov

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  1. S. Sh. Kozhegel’dinov
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Correspondence to S. Sh. Kozhegel’dinov.

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Original Russian Text © S. Sh. Kozhegel’dinov, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 3, pp. 365–377.

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Kozhegel’dinov, S.S. The al-husayn equation x 4 + y 2 = z 2 . Math Notes 89, 349–360 (2011). https://doi.org/10.1134/S0001434611030060

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