It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld A-module of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of K. Moreover, we show that within an L-iso...

Let $$N=2n^2-1$$ or $$N=n^2+n-1$$ , for any $$n\ge 2$$ . $$M=\frac{N-1}{2}$$ We construct families of prime knots with Jones polynomials $$(-1)^M\sum _{k=-M}^{M} (-1)^kt^k$$ Such have Mahler measure equal to 1. If N is prime, these are cyclotomic $$\Phi _{2N}(t)$$ up some shift in the powers t. Otherwise, they products such polynomials, including In particular, all roots unity $$\zeta _{2N}$$ o...

Definition 2.1. Let R be a ring. A left R-module is an abelian group (M,+) together with an R-action · : R ×M → M which satisfies M1 1 ·m = m for all m ∈ M; M2 r(s ·m) = (r s) ·m for all r, s ∈ R and m ∈ M; M3 r · (m + n) = (r ·m) + (s · n) A submodule N of M is a subset which is closed under + and satisfies r · n ∈ N for all r ∈ R and n ∈ N. A right R-module is defined similarly, but modules w...

Let A be a subset of {1, 2, . . . , n} such that the sum of no two distinct elements of A is a prime number. Such a subset is called a prime-sumset-free subset of {1, 2, . . . , n}. A prime-sumset-free subset is called an extremal prime-sumset-free subset of {1, 2, . . . , n} if A ∪ {a} is not a prime-sumset-free subset for any a ∈ {1, 2, . . . , n} \ A. We prove that if n ≥ 10 then there is no...