一类不定方程对 Lucas 序列的应用

APPLICATIONS OF DIOPHANTINE EQUATIONS TO LUCAS'SEQUENCES

1 引言设 D 不是平方数,α=a+b D1/2∈Q(D1/2),这里 a,b（?）0是有理整数。柯召和孙琦以及本文作者曾分别证明了不定方程α4m+β4m=2t2,N（α）=αβ （1）当 N（α）=1以及 N（α）=-1时无正整数解 m,t。最近,孙琦考虑了 N（α）=k,|k|】1的情形,他证明了 k 满足以下三个条件时方程（1）无正整数解:1)k 无平方因子;2)k 的任一个素因子 p 满足 p（?）Δ,这里Δ是 Q(D1/2)的基数;

In this paper,applications of Diophantine equationsa^(2m)+β^(2m)=2t^2,N(a)=αβ (1)and a^n+β~n=2t^2,N(α)=αβ (2)to Lucas'sequencesv_0=2,v_1=2a,v_(n+2)=2av_(n+1)-kv_n (3)are discussed.Where α=a+b D^(1/2),β=a-b D^(1/2)∈Q(D^(1/2)) and N(α)=k.Theorem 1 If there exists prime q of the forms 8f+3 or 8f+5 satisfyq‖k,q(?)Δ A,where Δ is discriminant of the field Q(D^(1/2)),then (1) has no posi-tive integer solutions m,t.Theorem 2 If α>0 and is not a perfect square,k satisfy(a) k≡1(mod4) is a square-free integer which is divisible by primes offorms 8f+3 or 8f+5;(b)p|k(?) p(?)Δ,where p is a prime and Δ is discriminant of Q(D^(1/2))Then (2) has no positive integer solutions n,t.Theorem 3 Let α>0 be not a perfect square,a^2-k=Db^2,where D is asquare-free.If k≡1(mod4)is a square-free integer which is divisible byprimes of forms 8f+3 or 8f+5,and p|k(?)p(?)Δ,where p is a prime and Δ isdiscriminant of Q(D^(1/2)),then Lucas'sequences (3) satisfies (v_n)/2 are not theperfect square,except (v_0)/2=1.