2015/02/25 by Maciej Gawron, Gawron, Maciej, Maciej Ulas +1
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1502.07307
openalex publication_date 2015/02/25 · openalex created_date 2022/08/16 · openalex updated_date 2026/07/28
In this paper we investigate Diophantine equations of the form\nT2=G(\X), ; \X=(X1,\…,Xm), where m=3 or\nm=4 and G is specific homogenous quintic form. First, we prove that if\nF(x,y,z)=x2+y2+az2+bxy+cyz+dxz\∈ Z[x,y,z] and (b-2,4a-d2,d)\≠\n(0,0,0), then the Diophantine equation t2=nxyzF(x,y,z) has solution in\npolynomials x, y, z, t with integer coefficients, without polynomial common\nfactor of positive degree. In case a=d=0, b=2 we prove that there are\ninfinitely many primitive integer solutions of the Diophantine equation under\nconsideration. As an application of our result we prove that for each\nn\∈ Q\∖ 0 the Diophantine equation nT2=n(X15+X25+X35+X45) has a solution in\nco-prime (non-homogenous) polynomials in two variables with integer\ncoefficients. We also present a method which sometimes allow us to prove the\nexistence of primitive integers solutions of more general quintic Diophantine\nequation of the form T2=aX15+bX25+cX35+dX45, where a, b, c,\nd\∈ Z. In particular, we prove that for each m, n\∈ Z\∖ 0 , the\nDiophantine equation nT2=m(X15-X25)+n2(X35-X45) has a solution in\npolynomials which are co-prime over Z[t]. Moreover, we show how modification\nof the presented method can be used in order to prove that for each\nn\∈ Q\∖ 0 , the Diophantine equation nt2=n(X15+X25-2X35) has a solution in polynomials\nwhich are co-prime over Z[t].\n