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 Advances in Pure Mathematics, 2011, 1, 133-135 doi:10.4236/apm.2011.14026 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Gauss’ Problem, Negative Pell’s Equation and Odd Graphs Aleksander Grytczuk Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Zielona Góra, Poland Department of Mathematics and Applications, Jan Paweł II Western Higher School of Commerce and International Finance, Zielona Góra, Poland E-mail: A.Grytczuk@wmie.uz.zgora.pl Received March 30, 2011; revised May 4, 2011; accepted May 15, 2011 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 =KQd, generated by Fermat’s numbers with 121==2mmdF 1,2m 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 [1]. Keywords: Fermat Numbers, Class-Number Gauss’ Problem, Odd Graphs 1. Introduction Let sZ be the set of all square-free positive integers. Moreover, let denote the class-number of real hdquadratic field =,.sKQddZIt is well-known (Cf. [2]) that the condition is equivalent to unique factorization in the ring =1hdKR of the algebraic integers of the field .KThe difficult and still open problem posed by Gauss concern of the existence infinitely many sdZ such that It is known that if hd =1.=1hd in the field =KQd then =, 2, dpq,qrwhere ,pq and are primes such that r34od.qr m This result has been proved by Hasse in the paper [3].Another proof of this result has been presented by Szymiczek [4]. In the paper [5] we give some arithmetic description of the set sZ and as consequence we also obtained this result.Many others interesting and impor- tant results concerning factorization problem have been given in the papers [1,6-7,9-14]. In this paper we prove of the following two theorems: Theorem 1: Let be the 121==2 1,mmdF m2Fermat’s number and let be the class- 1=mhd hF number of the real quadratic field =.KQdThen we have 1mmMhF (*) where is the Mersenne number. =2 1mmMTheorem 2: Let sdZ awnd here is the number of all distinct prime divisors of f there is primitive Pythagorean triple 2,d d.I,, and positive relatively prime integers a, b such that 22=,da b =1ab  (**) then >1hd (***) 2. Basic Lemmas Lemma 1: (H. W. Lu, [13]). Let 2=4 1ndk where are positive integers such that ,kn >1k and >1n. Then we have 0modhd n (2.1) where hd denote of the class-number of the field =.KQd Lemma 2: (A. Grytczuk, F. Luca, M. Wójtowicz, [15]). The negative Pell equation 22=1xdy (2.2) has a solution in positive integers x, y if and only if there is a primitive Pythagorean triple ,, and positive relatively prime integers a, b such that 22=,da b =1ab  (2.3) 134 A. GRYTCZUK Lemma 3: (A. Grytczuk, J. Grytczuk [5]). Let sZ be the set of all square-free integers and let p, pj for be primes. Moreover, let =1,2,...,jk1=;=...1,1 mod4skAdZdpppk p =;=2,1modsBdZd pp42,k 12=;=2...,3mod4,= 1, 2,...,skjCdZd pppkpjk 12=; =...,3,3mod4,= 1, 2,...,skjDdZdpp pkpj and hd be the class-number of the real quadratic number field =.KQd If then ,dABCD>1.hd 3. Proof of the Theorem 1 For the proof of (*) in the Theorem 1 we use Lemma 1. First we note that is positive integer, hence we put and let =2n mom this fact we obtain >1km=2k 1>1, soFr2. 22 1122 21=41=2 21=21=mmnmdk F  (3.1) By (2.1) of Lemma 1 it follows that nh d (3.2) Since is m-th Mersenne number then from (3.1) and (3.2) follows that (*) is true . =2 1=mmnMThe proof of the Theorem 1 is complete. From the Theorem 1 and some property of Fermat numbers follows the following Corollary: Corolla ry1: For each positive integer we have 1k011 21... kkFFF hF (3.3) where are Fermat numbers. 2=2 1jjFProof. We use well-known (see, Cf. [16]) the following identity: 01 1=... 2;kkFFF F  (3.4) 1kSince then from (3.4) we obtain 2=2 1kkF2k01 121= ...kFFF (3.5) Since thus by (3.5) and (*) it follows 2221=kkM,the divisibility (3.3) and the proof of Corollary 1 is finished.  4. Proof of the Theorem 2 Since sdZ and 2d then by the assumption of the Theorem 2 and Lemma 2 it follows that negative Pell’s equation (2.2) has a solution in positive relatively prime integers x, y. From the assumption that 2d it follows that there is a prime such that p.pd By (2.2) it follows that 21= 2xdy (4.1) From the well-known properties of divisibility relation and (4.1) we get 21px  (4.2) By the relation (4.2) it follows that 21xmodp (4.3) From (4.3) we see that 1 is a quadratic residue and consequently we have 1=1p (4.4) On the other hand from the property of Legendre’s symbol (Cf. [17], p. 342) we have 121=1pp (4.5) By (4.4) and (4.5) it follows that the prime number is the form: p=4 1.pkSince then we see that 2ddA or dB and by Lemma 3 it follows that >1.hd The proof of the Theorem 2 is complete.  5. Connections Negative Pell’s Equation with Gauss’ Problem and Graphs Theory Let P denote the set of all primes and let =1<,andthen 14sDdZifpPpdpmod (5.1) Let nDD, =nd; and 1,pq P,pqD. Then the following relation has been defined by Cremona and Odoni in the paper [1]: R11,pqRD Dp q and 32 3pxmodq for some xZ (5.2) Let ndD and 12=...ndpp p ; , 12<< ... 1.hd 6. References [1] J. E. Cremona and R. W. K. Odoni, “Some Density Re-sults for Negative Pell Equations: An Application on Graphs Theory,” Journal of the London Mathematical Society, Vol. 39, No. 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 Mathe-matica 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,” Acta Mathematica Sinica, Vol. 21, No. 5, 2005, pp. 1107-1112. HUdoi:10.1007/s10114-005-0544-2U [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 Quad-ratic Residues,” Glasgow Mathematical Journal, Vol. 17, No. 1, 1976, pp. 47-52. doi:10.1017/S0017089500002718U [9] S. Herz, “Construction of Class Fields,” Seminar on Complex Multiplication, Lectures Notes in Mathematics, Vol. 21, 1966, pp. VII-1-VII-21. [10] M. H. Le, “Divisibility of the Class Number of the Real Quadratic Field 214 ,kQa” 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,” Dis-cussion in Mathematics, Vol. 13, 1993, pp. 69-73. [17] W. Sierpinski, “Elementary Theory of Numbers,” PWN, Warszawa, 1987.