In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer D, we show how to (practically) solve Diophantine equations of the form x −Dy = 1 in integers x, y and n ≥ 2. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms...

Let gn1,n2 be the number of standard Young tableau of truncated shifted shape with n1 rows and n2 boxes in each row. By using the integral method this paper derives the recurrence relations of g3,n, gn,4 and gn,5 respectively. Specifically, gn,4 is the (2n− 1)-st Pell number.

Abstract. The m, n Turk’s Head Knot, THK(m, n), is an “alternating (m, n) torus knot.” We prove the Harary-Kauffman conjecture for all THK(m, n) except for the case where m ≥ 5 is odd and n ≥ 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m, 2) when Pm ...

In [W2] West conjectured that there are 2(3n)!/((n+1)!(2n+1)!) two-stack sortable permutations on n letters. This conjecture was proved analytically by Zeilberger in [Z]. Later, Dulucq, Gire, and Guibert [DGG] gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on n letters avoiding (or containing ex...

Abstract Let $$d \ge 2$$ d ≥ 2 be an integer which is not a square. We show that if $$(F_n)_{n\ge 0}$$ ( F n ) 0 the Fibonacci sequence and $$(X_...