The Prime Factors of Wendt's Binomial Circulant Determinant

On Wendt's determinant

Wendt’s determinant of order m is the circulant determinant Wm whose (i, j)-th entry is the binomial coefficient ( m |i−j| ) , for 1 ≤ i, j ≤ m. We give a formula for Wm, when m is even not divisible by 6, in terms of the discriminant of a polynomial Tm+1, with rational coefficients, associated to (X + 1)m+1 −Xm+1 − 1. In particular, when m = p − 1 where p is a prime ≡ −1 (mod 6), this yields a...

On Wendt's Determinant and Sophie Germain's Theorem

Research supported by the Natural Sciences and Engineering Research Council (Canada) and Fonds pour la Formation de Chercheurs et l'Aide a la Recherche (Quebec). Some results in section 3 of this work are taken from Jha's Ph.D. thesis [Jha 1992]. After a brief review of partial results regarding Case I of Fermat’s Last Theorem, we discuss the relationship between the number of points on Fermat’...

Upper bounds for the prime divisors of Wendt's determinant

Let c ≥ 2 be an even integer, (3, c) = 1. The resultant Wc of the polynomials tc − 1 and (1 + t)c − 1 is known as Wendt’s determinant of order c. We prove that among the prime divisors q of Wc only those which divide 2c−1 or Lc/2 can be larger than θc/4, where θ = 2.2487338 and Ln is the nth Lucas number, except when c = 20 and q = 61. Using this estimate we derive criteria for the nonsolvabili...

Fleck’s binomial congruence using circulant matrices

Spectral norms of circulant-type matrices involving some well-known numbers

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skewcirculant and g-circulant matrices. The entries are product of binomial coefficients with Fibonacci numbers and Lucas numbers, respectively. We obtain identity estimations for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.