Research On The Problems Of From Numbers In Two Recurrent Sequences Arising In The Units Of Quadratic Field Q((?))

This paper discussses the problems of Pronic numbers,triangular numbers,pentago-nal numbers and heptagonal numbers in two recurrent sequences {Un} and {Vn},which are arising in the units Un+Vn √10=(19+6√10)n of quadratic field Q(√10)and 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 equations.The detailed results are as follows:Theorem3.1 There has no Pronic number in {Un}.Theorem3.5 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}.Theorem4.6 Vn is a triangular number if and only if n=1 in{Vn}.Theorem5.3 Un is a generalized pentagonal number or pentagonal number if and only if n=0 in{Un}.Theorem5.6 Vn is a generalized pentagonal number if and only if n=0 in{Vn}.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 {Vn}.Theorem7.1 The Diophantine equations x~2(x+1)~2-10y2=1 has no integral solution.Theorem7.2 The integral solutions of the Diophantine equation x~2-10y2(y+1)~2=1 such that x>0 are(1,0),(1,-1),(19,2),(19,-3).Theorem7.3 The integral solutions of the Diophantine equation x~2(x+1)~2-40y2=4 are(1,0),(-2,0).Theorem7.4 The integral solutions of the Diophantine equation 2x~2-5y2(y+1)~2=2 such that x>0 are(1,0),(1,-1),(19,3),(19,-4).Theorem7.5 The integral solutions of the Diophantine equation x~2(3x-1)~2-40y2=4 such that x≥0 is (1,0).Theorem7.6 The integral solutions of the Diophantine equation 2x~2-5y2(3y-1)~2=2 such that x>0 is(1,0).Theorem7.7 The integral solutions of the Diophantine equation x~2(5x-3)~2-40y2=4 is (1,0).Theorem7.8 The integral solutions of the Diophantine equation 2x~2-5y2(5y-3)~2=2 such that x> 0 is (1,0).