Pairs of IntegersDate: 08/16/97 at 07:08:39 From: pal sandip Subject: Number theory Show that there are infinitely many pairs of integers(x,y) such that x|y**2+m and y|x**2+m where m is any chosen integer. Moreover, gcd(x,y) = 1. Date: 08/21/97 at 09:01:46 From: Doctor Rob Subject: Re: Number theory This is certainly false if m = 0, since then x|y^2 and (x,y) = 1 => x = 1 or -1, and similarly y = 1 or -1, so there are just four pairs. For m = -a^2, this is easy: (y-a,y) and (y+a,y) with (y,a) = 1 are two infinite families, and (a,a*z) is another (although not relatively prime unless a = 1). For m = 1, the sequence of values which works is: (1,2), (2,5), (5,13), (13,34), (34,89), (89,233), (233,610), ... which are the Fibonacci numbers (F[2*k-1], F[2*k+1]). The appropriate identity is F[2*j-1]*F[2*j+3] = F[2*j+1]^2 + 1 applied with j = k and with j = k - 1. The recursion satisfied by the x-values (and by the y-values) is x[n+1] = 3*x[n] - x[n-1]. For m = 2, the sequence of values which works is: (1,3), (3,11), (11,41), (41,153), (153,571), (571,2131), ... The recursion satisfied by the x-values (and by the y-values) is x[n+1] = 4*x[n] - x[n-1]. There is the identity x[j-1]*x[j+1] = x[j]^2 + 2, which is applied with j = n and j = n-1. For m = 3, there are two infinite sequences of the same sort: (1,2), (2,7), (7,26), (26,97), (97, 362), (362, 1351), ... (1,4), (4,19), (19,91), (91,436), (436,2089), ... The first satisfies x[n+1] = 4*x[n] - x[n-1], which we treated in the preceding paragraph, and the second satisfies x[n+1] = 5*x[n] - x[n- 1]. Presumably there is an associated identity which works here, too. For m = 4, there is one sequence: (1,5), (5,29), (29,169), (169,985), ... which satisfies x[n+1] = 6*x[n] - x[n-1]. Presumably there is an associated identity which works here, too. Do you detect a pattern? -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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