Gauss’ Problem, Negative Pell’s Equation and Odd Graphs
Author(s)
Aleksander Grytczuk
ABSTRACT
In this paper we present some results connected with still open problem of Gauss, negative Pell’s equation and some type graphs.In particular we prove in the Theorem 1 that all real quadratic fields K=Q( ) , generated by Fermat’s numbers with d=Fm+1=22m+1+1,m≥2, have not unique factorization. Theorem 2 give a connection of the Gauss problem with primitive Pythagorean triples. Moreover, in final part of our paper we indicate on some connections of the Gauss problem with odd graphs investigated by Cremona and Odoni in the papper [5].
In this paper we present some results connected with still open problem of Gauss, negative Pell’s equation and some type graphs.In particular we prove in the Theorem 1 that all real quadratic fields K=Q( ) , generated by Fermat’s numbers with d=Fm+1=22m+1+1,m≥2, have not unique factorization. Theorem 2 give a connection of the Gauss problem with primitive Pythagorean triples. Moreover, in final part of our paper we indicate on some connections of the Gauss problem with odd graphs investigated by Cremona and Odoni in the papper [5].
Cite this paper
nullA. Grytczuk, "Gauss’ Problem, Negative Pell’s Equation and Odd Graphs," Advances in Pure Mathematics, Vol. 1 No. 4, 2011, pp. 133-135. doi: 10.4236/apm.2011.14026.
nullA. Grytczuk, "Gauss’ Problem, Negative Pell’s Equation and Odd Graphs," Advances in Pure Mathematics, Vol. 1 No. 4, 2011, pp. 133-135. doi: 10.4236/apm.2011.14026.
References
[1] J. E. Cremona and R. W. K. Odoni, “Some density results for negative Pell equations:an application on graphs theory,” Journal of the London Mathematical Society, Vol. 39, Vol. 1, 1993, pp. 16-28. doi:10.1112/jlms/s2-39.1.16 [2] W. Narkiewicz, “Elementary and Analytic Theory of Algebraic Integers,” PWN, Warszawa, 1990. [3] H. Hasse, “Uber Mehrklassige Uaber Eigen Schlechtige Reel-Quadratische Zahlkorper,” Elementary Mathematics, Vol. 20, 1965, pp. 49-59. [4] K. Szymiczek, “Knebush-Milnor Exact Sequence and Parity of Class Numbers,” Ostraviensis Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 4, 1996, pp. 83-95. [5] A. Grytczuk and J. Grytczuk, “Some Results Connected with Class Number in Real Quadratic Fields,” Monatshefte für Mathematik, Vol. 21, No. 4, 2005, pp. 1107-1112. doi:10.1007/s10114-005-0544-2 [6] A. Biro, “Chowla’s Conjecture,” Acta Arithmetica, Vol. 107, No. 2, 2003, pp. 179-194. doi:10.4064/aa107-2-5 [7] Z. Cao and X. Dong, “Diophantine Equations and Class Numbers of Real Quadratic Fields,” Acta Arithmetica, Vol. 97, No. 4, 2001, pp. 313-328. [8] S. Chowla and J. Friedlander, “Class Numbers and Quadratic Residues,” Glasgow Mathematical Journal, Vol. 17, No. 1, 1976, pp. 47-52. doi:10.1017/S0017089500002718 [9] S. Herz, “Construction of Class Fields,” Seminar on Complex Multiplication, Lectures Notes in Mathematics, Vol. 21, 1966, VII-1-VII-21. [10] M. H. Le, “Divisibility of the Class Number of the Real Quadratic Field ” Acta Mathematica Sinica, Vol. 33, 1990, pp. 565-574. [11] H. W. Lu, “The Divisibility of the Class Number of Some Real Quadratic Fields,” Acta Mathematica Sinica, Vol. 28, 1985, pp. 56-762. [12] R. Mollin and H. C.Williams, “A conjecture of S. Chowla via Generalised Riemann Hypothesis,” Proceedings of the American Mathematical Society, Vol. 102, 1988, pp. 794-796. doi:10.1090/S0002-9939-1988-0934844-9 [13] R. Mollin, “Quadratics,” CRC Press, Boca Raton, 1995. [14] P. Z. Yuan, “The Divisibility of the Class Numbers of Real Quadratic Fields,” Acta Mathematica Sinica, Vol. 41, 1998, pp. 525-530. [15] A. Grytczuk, F. Luca and M. Wójtowicz, “The Negative Pell Equation and Pythagorean Triples,” Proceedings of the Japan Academy, Vol. 76, No. 6, 2000, pp. 91-94. doi:10.3792/pjaa.76.91 [16] A. Grytczuk, “Some remarks on Fermat numbers,” Discussion in Mathematics, Vol. 13, 1993, pp. 69-73. [17] W. Sierpinski, “Elementary Theory of Numbers,” PWN, Warszawa, 1987.
[1] J. E. Cremona and R. W. K. Odoni, “Some density results for negative Pell equations:an application on graphs theory,” Journal of the London Mathematical Society, Vol. 39, Vol. 1, 1993, pp. 16-28. doi:10.1112/jlms/s2-39.1.16 [2] W. Narkiewicz, “Elementary and Analytic Theory of Algebraic Integers,” PWN, Warszawa, 1990. [3] H. Hasse, “Uber Mehrklassige Uaber Eigen Schlechtige Reel-Quadratische Zahlkorper,” Elementary Mathematics, Vol. 20, 1965, pp. 49-59. [4] K. Szymiczek, “Knebush-Milnor Exact Sequence and Parity of Class Numbers,” Ostraviensis Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 4, 1996, pp. 83-95. [5] A. Grytczuk and J. Grytczuk, “Some Results Connected with Class Number in Real Quadratic Fields,” Monatshefte für Mathematik, Vol. 21, No. 4, 2005, pp. 1107-1112. doi:10.1007/s10114-005-0544-2 [6] A. Biro, “Chowla’s Conjecture,” Acta Arithmetica, Vol. 107, No. 2, 2003, pp. 179-194. doi:10.4064/aa107-2-5 [7] Z. Cao and X. Dong, “Diophantine Equations and Class Numbers of Real Quadratic Fields,” Acta Arithmetica, Vol. 97, No. 4, 2001, pp. 313-328. [8] S. Chowla and J. Friedlander, “Class Numbers and Quadratic Residues,” Glasgow Mathematical Journal, Vol. 17, No. 1, 1976, pp. 47-52. doi:10.1017/S0017089500002718 [9] S. Herz, “Construction of Class Fields,” Seminar on Complex Multiplication, Lectures Notes in Mathematics, Vol. 21, 1966, VII-1-VII-21. [10] M. H. Le, “Divisibility of the Class Number of the Real Quadratic Field ” Acta Mathematica Sinica, Vol. 33, 1990, pp. 565-574. [11] H. W. Lu, “The Divisibility of the Class Number of Some Real Quadratic Fields,” Acta Mathematica Sinica, Vol. 28, 1985, pp. 56-762. [12] R. Mollin and H. C.Williams, “A conjecture of S. Chowla via Generalised Riemann Hypothesis,” Proceedings of the American Mathematical Society, Vol. 102, 1988, pp. 794-796. doi:10.1090/S0002-9939-1988-0934844-9 [13] R. Mollin, “Quadratics,” CRC Press, Boca Raton, 1995. [14] P. Z. Yuan, “The Divisibility of the Class Numbers of Real Quadratic Fields,” Acta Mathematica Sinica, Vol. 41, 1998, pp. 525-530. [15] A. Grytczuk, F. Luca and M. Wójtowicz, “The Negative Pell Equation and Pythagorean Triples,” Proceedings of the Japan Academy, Vol. 76, No. 6, 2000, pp. 91-94. doi:10.3792/pjaa.76.91 [16] A. Grytczuk, “Some remarks on Fermat numbers,” Discussion in Mathematics, Vol. 13, 1993, pp. 69-73. [17] W. Sierpinski, “Elementary Theory of Numbers,” PWN, Warszawa, 1987.