A Generalization of the Ramanujan–nagell Equation

Abstract

We shall show that, for any positive integer D > 0 and any primes p 1 , p 2 not dividing D, the diophantine equation x 2 + D = 2 s p k 1 p l 2 has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}. IntroductionIt is known that the equation x 2 + 7 = 2 n has five solutions, as conjectured by Ramanujan and shown by Nagell [26] and other authors. According to this history, this diophantine equation has been called the Ramanujan-Nagell equation and several authors have studied various analogues.Apéry [1] showed that, for each integer D > 0 and prime p, the equation x 2 + D = p n has at most two solutions unless (p, D) = (2, 7) and, for any odd prime p, the equation x 2 +D = 4p n , which is equivalent to y 2 +y +(D +1)/4 = p n with y odd, also has at most two solutions. Beukers [5] showed that, if D > 0 and x 2 + D = 2 n has two solutions, then D = 23 or D = 2 k − 1 for some k > 3 and also gave an effective upper bound: if w = x 2 + D = 2 n with D = 0, then w < 2 435 |D| 10 . Further generalizations have been made by Le [18][19][20], Skinner [29] andBender and Herzberg [2] to prove that, for any given integers A, B, s, p with gcd(A, B) = 1, s ∈ {0, 2} and p prime, Ax 2 + B = 2 s p n has at most two solutions except 2x 2 + 1 = 3 k , 3x 2 + 5 = 2 k , x 2 + 11 = 4 × 3 k , x 2 + 19 = 4 × 5 k with three solutions and the Ramanujan-Nagell one x 2 + 7 = 2 k with five solutions.Bender and Herzberg [2] also found some necessary conditions for the equation D 1 x 2 + D 2 = 2 s a n with D 1 > 0, D 2 > 0, gcd(D 1 , D 2 ) = gcd(D 1 D 2 , k) = 1, s ∈ {0, 2} to have more than 2 ω(a) solutions. With the aid of the primitive divisor theorem of Bilu, Hanrot and Voutier [7] concerning Lucas and Lehmer sequences, Bugeaud and Shorey[10] determined all cases D 1 x 2 + D 2 = 2 m a n with D 1 > * 2010 Mathematics Subject Classification: 11D61 (Primary), 11D45 (Secondary).† Key words and phrases: Exponential diophantine equation, Ramanujan-Nagell equation, values of quadratic polynomials.

Three-term Machin-type formulae

“…Lemma 2.1. Let x 1 , x 2 , x 3 , y 1 , y 2 , y 3 and r be nonzero integers with x 1 , x 2 , x 3 > 1 satisfying (13) and m 1 , m 2 , s i , k i , l i (i = 1, 2, 3) be corresponding integers satisfying (15).…”

“…Let x 1 , x 2 , x 3 , y 1 , y 2 , y 3 and r be integers with x 1 , x 2 , x 3 > 1, r = 0 satisfying (13) and m 1 , m 2 , s i , k i , l i (i = 1, 2, 3) be corresponding integers, η 1 , η 2 be gaussian integers satisfying (14) and (15). We write K = max k i and L = max l i .…”

“…Evertse [2] proved that (16) has at most 3 • 7 4n+6 solutions. In the case n = 2, the author [15] reduced Evertse's bound 3 • 7 14 to 63.…”