Research On The Problems Of From Numbers In Two Recurrent Sequences Arising In The Units Of Quadratic Field Q(6^1/2)

h this paper,we shall discuss the problem of basic form numbers(Pronic numbers,tri-angular numbers,pentagonal numbers and heptagonal numbers)in two recurrent sequences {Un}and{Vn),which arising in the unitsUn+Vn(?)3=(5+2(?)6)n of quadratic field Q(?)6. We solve the problems completely and find out all four kinds of form numbers in{Un}and {Vn}.As applications,we obtain all integer solutions of eight related Diphantine equations. The detailed results are as follows:Theorem3.1.There has no Pronic number in{Un}.Theorem3.7.Vn is a Pronic number if and only if n=1,2in{Vn}.Theorem4.1.Unis a triangular number if and only ifn=0in{Un}.Theorem4.5.There has no triangular number in{Vn}.Theorem5.4.Un is a generalized pentagonal number or pentagonal number if and only if n=0,±1in{Un}.Theorem5.8.Vn is a generalized pentagonal number if and only if n=0,1,in which there has no pentagonal number in{Vn}.Theorem6.4.Un is a heptagonal number if and only if n=0in{Un}.Theorem6.8.There has no heptagonal number in{Vn}.Theorem7.1.The Diphantine equation X2(x+1)2-6y2=1has no integral solution.Theorem7.2.The integral solutions of the Diphantine equation x2-6y2(y+1)2=1such that x>0are(1,0),(1,一1),(5,一2),(5,1),(49,-5),(49,4).Theorem7.3.The integral solutions of the Diphantine equation x2(x+1)2-24y2=4are(1,0),(-2,0)_Theorem7.4.The integral solutions of the Diphantine equation2x2-3y2(x+1)2=2such that x>0are(1,0),(1,一1).Theorem7.5.The integral solutions of the Diphantine equation x2(3x+1)2-24y2=4 Such that y≥0are(1,0),(2,2).Theorem7.6.The integral solutions of the Diphantine equation2x2-3y2(3y-1)2=2such thatx>0are(1,0),(5,一1).Theorem7.7.The integral solutions of the Diphantine equation x2(5x-3)2-24y2=4is(1,0).Theorem7.8.The integral solutions of the Diphantine equation2x2-3y2(5y-3)2=2such that x>0is(1,0).