Research On The Problems Of Form Numbers In Two Recurrent Sequences Arising In The Units Of Quadratic Field Q(3～(1/3))

In this paper we shall discuss the problems of basic form numbers—Pronic numbers , triangular numbers and pentagonal numbers in two recurrent sequences {U_n} and {V_n}, which arising in the units U_n + V_n3 =(2+3)ⁿ of quadratic field Q (3). We solve the problems completely and find out all three kinds of form numbers in {U_n} and {V_n}. As applications, we obtain all integer solutions of six related Diophantine equations, as well as give an elementary solution of another Diophantine equation arising from combinatorial designs. The detailed results are as follows:Theorem 1 4Un+1 is a perfect square number if and only if n=±1, so that U±1=2 is the only Pronic number in {U_n}.Theorem 2 The only Pronic numbers in {V_n} are V0=0 and V4=56.Theorem 3 The only triangular number in {U_n} is U0=1.Theorem 4 The only triangular numbers in {V_n} are V1=1, V3=15 and V6=780.Theorem 5 Un is a generalized pentagonal number if and only if n=0,±1,±2,±3, in which U₀=1 is the only pentagonal number.Theorem 6 Vn is a generalized pentagonal number if and only if n=0, 1, 3, in which V₁=1 is the only pentagonal number.Theorem 7 The only integral solutions of the Diophantine equation x² ( x+ 1)²-3y²=1 such that y>0 are (x, y)=(-2, 1), (1, 1).Theorem 8 The only integral solutions of the Diophantine equation x² - 3 y²(y+1)²=1 such that x>0 are (x, y)=(1, -1), (1, 0), (97, -8), (97, 7).Theorem 9 The only integral solutions of the Diophantine equation x² ( x+ 1)²-12y²=4 are (x, y)=(-2, 0), (1, 0).Theorem 10 The only integral solutions of the Diophantine equation 4 x 2 - 3y²(y+1)²=4 such that x>0 are (x, y)=(2, -2), (2, 1), (26, -6), (26, 5), (1351, -40), (1351, 39).Theorem 11 The only integral solutions of the Diophantine equation x² (3 x- 1)²-12y²=4 such that y 0 are (x, y)=(1, 0), (-1, 1), (-2, 4), (-4, 15).Theorem 12 The only integral solutions of the Diophantine equation 4 x 2 - 3y²(3y-1)²=4 such that x>0 are (x, y)=(1, 0), (2, 1), (26, -3).Theorem 13 The only positive integral solutions of the Diophantine equation 3x4 -4y⁴ -2x² +12y² -9=0 are (x, y)=(1, 1) and (3, 3).