Research On The Problems Of From Numbers In Two Recurrent Sequences Arising In The Units Of Quadratic Field Q30^1/2

In this paper,we shall discuss the problem of basic form numbers(Pronic numbers,triangular numbers,pentagonal numbers and heptagonal numbers)in two recurrent se-quences {Un} and ?Vn},which are arising in the units Un + Vn(?)-(11+2(?))n of quadratic field Q((?)).We solves the problems completely and finds out all Pronic numbers,triangular numbers,pentagonal numbers and heptagonal numbers in ?Un} and{Vn}.As applications,we obtain all integer solutions of eight related Diophantine equa-tions.Theorem3.1 There has no Pronic number in {Un}.Theorem3.6 Vn is a Pronic number if and only if n = 1 in {Vn}.Theorem4.1 Un is a triangular number if and only if n = 0 in {Un}.Theorem4.5 There has no triangular number in {Vn}.Theorem5.3 Un is a generalized pentagonal number or pentagonal number if and only if n = 0 in {Un}.Theorem5.7 Vn is a generalized pentagonal number if and only if n = 0,1 in{Un}.which there has no pentagonal number in {Vn}.Theorem6.4 Un is a heptagonal number if and only if n = 0 in {Un}.Theorem6.8 There has no heptagonal number in {Un}.Theorem7.1 The Diophantine equations x2(x+1)2-30y2 = 1 has no integral solution.Theorem7.2 The integral solutions of the Diophantine equation x2-30y2(y + 1)2 = 1 such that x>0 are(1,0),(1,-1),(11,1),(11,-2).Theorem7.3 The integral solutions of the Diophantine equation x2(x+1)2-120y2 = 4 are(1,0),(-2,0).Theorem7.4 The integral solutions of the Diophantine equation 2x2-15y2(y+1)2 = 2 such that x>0 are(1,0),(1,-1).Theorem7.5 The integral solutions of the Diophantine equation x2(3x-1)2-120y2 =4 such that x ? 0 is(1,0).Theorem7.6 The integral solutions of the Diophantine equation 2x2-15y2(3y-1)2 ?2 such that x>0 are(1,0),(11,-1).Theorem7.7 The integral solutions of the Diophantine equation x2(5x-3)2-120y2 =4 is(1,0).Theorem7.8 The integral solutions of the Diophantine equation 2x2-15y2(5y-3)2 =2 such that x>0 is(1,0).