All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 8 Next

Displaying 141 – 160 of 694

Showing per page

Perfect powers expressible as sums of two fifth or seventh powers

Sander R. Dahmen, Samir Siksek (2014)

Acta Arithmetica

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the non-existence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.

Perfect powers in the summatory function of the power tower

Florian Luca, Diego Marques (2010)

Journal de Théorie des Nombres de Bordeaux

Let ( a n ) n ≥ 1 be the sequence given by a 1 = 1 and a n = n a n - 1 for n ≥ 2 . In this paper, we show that the only solution of the equation a 1 + ⋯ + a n = m l is in positive integers l > 1 , m and n is m = n = 1 .

Perfect unary forms over real quadratic fields

Dan Yasaki (2013)

Journal de Théorie des Nombres de Bordeaux

Let F = ℚ ( d ) be a real quadratic field with ring of integers 𝒪 . In this paper we analyze the number h d of GL 1 ( 𝒪 ) -orbits of homothety classes of perfect unary forms over F as a function of d . We compute h d exactly for square-free d ≤ 200000 . By relating perfect forms to continued fractions, we give bounds on h d and address some questions raised by Watanabe, Yano, and Hayashi.

Currently displaying 141 – 160 of 694