Let (Tn) n≥0 denote the Tripell sequence, defined by the linear recurrence Tn = 2Tn−1 + Tn−2 + Tn−3 for n ≥ 3 with T0 = 0 , T1 = 1 and T2 = 2 as initial conditions. In this paper, we study the 2-adic and 3-adic valuation of the Tripell sequence and, as an application, we determine all Tripell numbers which are factorials.
In this paper, we study the variant of the Brocard-Ramanujan diophantine equation m! + 1 = u 2 , where u is a member of a sequence of positive integers. Under some technical conditions on the sequence, we prove that this equation has at most finitely many solutions in positive integers m and u. As an application, we completely solve this equation when u is a Tripell number. The Tripell numbers are defined by the recurrence relation Tn = 2Tn−1 + Tn−2 + Tn−3 for n ≥ 3, with T0 = 0, T1 = 1 and T2 = 2 as initial conditions. This is known as the Brocard-Ramanujan diophantine equation, and it is still an open problem (see [11]). It is expected that the only solutions are (m, n) = (4, 5), (5, 11), (7, 71). Computations by Berndt and Galway [1] showed that there are no other solution in the range n < 10 9 . In 1993, Overholt [20] proved that the weak form of Szpiro's conjecture implies that equation (1) has
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