New methods for obtaining new families of congruent numbers

Hamid Reza Abdolmalki and Farzali Izadi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 14—24
DOI: 10.7546/nntdm.2019.25.1.14-24
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Authors and affiliations

Hamid Reza Abdolmalki
Department of Mathematics, Faculty of Science
Azarbaijan Shahid Madani University
Tabriz 53751-71379, Iran

Farzali Izadi
Department of Mathematics, Faculty of Science
Azarbaijan Shahid Madani University
Tabriz 53751-71379, Iran

Abstract

In this note, we introduce elementary methods for obtaining new families of congruent numbers (CNs). By our methods, we can produce other CNs when one or two CNs are given. Also, we use some of the Pell equations (PEs) for getting some families of CNs. Up to now, it is not exactly determined which prime numbers of the form p = 8k + 1 are CNs. Among other things, we also introduce two simple methods to find some CNs of the forms p ≡ 1 (mod 8) and 2p where p is a prime number. By non-CNs and our methods, we also obtain some Diophantine equations (especially of degree 4), which have no positive solutions. In the end, we obtain a result on Heron triangles.

Keywords

  • Congruent numbers
  • Elliptic curves
  • Rank
  • Pythagorean triples
  • Pell equations
  • Heron triangles

2010 Mathematics Subject Classification

  • 11G05
  • 14H52
  • 14G05
  • 11E16
  • 11D09

References

  1. Adler, A., & Cloury, J. E. (1995). The Theory of Numbers: A Text and Source Book of Problems. Boston.: Jones and Bartlett Publishers.
  2. Alter, R., & Curtz, T. B. (1974). A note on congruent numbers. Math. Comp. I, 303–305.
  3. Alter, R., & Curtz, T. B., & Kubota, K. K. (1972). Remarks and results on congruent numbers. Proc. Third Southeastern Conf. on Combinatorics, Graph Theory and Computing, pp. 27–35.
  4. Basor, E., & Hart, B. A Trillion Triangles. Available online: http://www.aimath. org/news/congruentnumbers/.
  5. Bastein, L., (1915). Nombers congruents. Intermediaire des Math, 22, 231–232.
  6. Beiler, A. H. (1966). “Chapter 14: The Eternal Triangle.” Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover.
  7. Birch, B. J. (1968). Diophantine analysis and modular functions, Proc., Bombay Colloq. Alg. Geom., 76, 35–42.
  8. Birch, B. J. (1970). Elliptic curves and modular functions, Symp. Math. 1st. Alta MAt., 4, 27–32.
  9. Carlson, J. R. (1970). Determination of Heronian triangles, Fibonacci Quarterly, 8, 499– 506.
  10. Dickson, L. E. (1934). History of the Theory of Numbers, Vol. II: Diophantine Analysis, G. E. Stechert Co., New York.
  11. Elkies, N. (1988). On A4 + B4 + C4 = D4. Mathematics of Computation, 51 (184), 825–835.
  12. Feng, K. (1996). Non-congruent numbers, odd graphs and the Birch–Swinnerton-Dyer conjecture, Acta Arith., 75, 71–83.
  13. Gerardin, A. (1915). Nombers congruent. Intermediaire des Math., 22, 52–53.
  14. Gross, B. H., & Zagier, D. B. (1986). Heegner points and derivatives of L-series, Invent. Math, 84 (2), 225–320.
  15. Heegner, K. (1952). Diophantine analysis und Modulfunctionen, Math. Z., 56, 227–253.
  16. Izadi, F. (2015). Coungruent number via the pell equation and its analogous counterpart, Notes on Number Theory and Discrete Mathematics, 21 (1), 70–78.
  17. Lagrange, J. (1974–1975). Nombres congruents et courbes elliptiques. Sm. Delange-Pisot- Poitou (Thorie des nombres), 16e anne, no. 16, 17 p.
  18. Lagrange, J., Construction dune table de nombres congruents. Manuscript, Reims.
  19. Lander L. J. & Parkin, T. R. (1966). Counterexamples to Euler’s conjecture on sums of like powers, Bull. Amer. Math. Soc., 72, p. 1079.
  20. Monsky, P. (1990). Mock Heegner points and congruent numbers, Math. Z., 204, 45–68.
  21. Nagell, T. (1929). Lanalyse indetermineede degre superieur. Gauthier-Villars, Paris, 39, 16–17.
  22. Nagell, T. (1981). Introduction to Number Theory, New York, Chelsea Publishing Company.
  23. Piezas III, T. (2010). A Collection of Algebraic Identities, Available online: https:// sites.google.com/site/tpiezas/Home.
  24. Sastryt, K. R. S. (2001). A Heron difference, Crux Math. & Math. Mayhem, 27, 22–26.
  25. Stephens, N. M. (1975). Congruence properties of congruent numbers, Bull. London Math. Soc., 7, 182–184.
  26. Tunnell, J. B. (1983). A classical Diophantine problem and modular forms of weight 3/2. Inv. Math.. 72 (2), 323–334.
  27. Washington, L. C. (2008). Elliptic Curves: Number Theory and Cryptography, Chapman- Hall.
  28. Yiu, P. (1998). Isosceles triangles equal in perimeter and area, Missouri J. Math. Sci., 10, 106–111.
  29. Yiu, P. (1998). Construction of indecomposable Heronian triangles, Rocky Mountain J. of Math., 28, 1189–1201.

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Cite this paper

Abdolmalki, H. R. & Izadi, F. (2019). New methods for obtaining new families of congruent numbers. Notes on Number Theory and Discrete Mathematics, 25(1), 14-24, doi: 10.7546/nntdm.2019.25.1.14-24.

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