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Gauss’ Problem, Negative Pell’s Equation and Odd Graphs

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Author(s): Aleksander Grytczuk

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 01;
Issue: 04;
Start page: 133;
Date: 2011;
Original page

Keywords: Fermat Numbers | Class-Number Gauss’ Problem | Odd Graphs

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].
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