Our goal is to prove the following theorem: Dirichlet’s Theorem: For any coprime a, b ∈ Z, there are infinitely many primes p such that p ≡ a (mod b). Although the statement of the theorem involves only integers, the simplest proof requires the use of complex numbers and Dirichlet L-series. Most of this paper will therefore be devoted to proving some basic properties of characters and L-series,...

Abstract The problem of synchronous Fractional Sample Rate Conversion (FSRC) of a digital signal by L/M, where L and M are coprime integers [3,13], is revisited. Based on a novel approach two different efficient causal block implementations of FSRC are concurrently derived and compared with each other. While the computational load of both structures, being performed in an LTI MIMO subsystem at ...

Let (a, b, c) be a primitive Pythagorean triple. Set $$a = m^2-n^2 $$ , $$b=2mn$$ and $$c=m^2+n^2$$ with m n positive coprime integers, $$ m>n $$m \not \equiv \pmod 2 . A famous conjecture of Jeśmanowicz asserts that the only integer solution to Diophantine equation $$a^x+b^y=c^z$$ is $$(x,y,z)=(2,2,2).$$ $$(x,y,z) \ne (2,2,2)$$ this called an exceptional solution. In note, we will prove ...

We consider the efficient solution of the coprime factorization based H∞ controller approximation problems by using frequency-weighted balancing related model reduction approaches. It is shown that for a class of frequency-weighted performance preserving coprime factor reduction as well as for a relative error coprime factor reduction method, the computation of the frequency-weighted controllab...

The study of core partitions has been very active in recent years, with the study of ps, tq-cores – partitions which are both sand t-cores – playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an ps, tq-core, when s and t are coprime positive integers, is 1 24ps ́1qpt ́1qps` t ́1q. Armstrong also conjectured that the same formula gives the...

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C∗ is prime. This definition, however, is not intrinsic it strongly depends on the base ring being a field...

This paper examines “Stoneham constants,” namely real numbers of the form αb,c = ∑ n≥1 1/(c nbc n ), for coprime integers b ≥ 2 and c ≥ 2. These are of interest because, according to previous studies, αb,c is known to be bnormal, meaning that every m-long string of base-b digits appears in the base-b expansion of the constant with precisely the limiting frequency b−m. So, for example, the const...

Let d ≥ 3 be an integer and p a prime coprime to d. Let Q and Qp be the algebraic closure of Q and Qp respectively. Let Zp be the ring of integers of Qp. Suppose f(x) is a degree-d polynomial in (Q ∩ Zp)[x]. Let Q(f) be the number field generated by coefficients of f in Q. Let P be a prime ideal in the ring of integers OQ(f) of Q(f) lying over p, with residue field Fq for some p-power q. Let A ...

The combinatorial sum of binomial coe cients  n i r (a) := X k⌘i (mod r) ✓ n k ◆ a k has been studied widely in combinatorial number theory, especially when a = 1 and a = 1. In this paper, we connect it with integer factorization for the first time. More precisely, given a composite n, we prove that for any a coprime to n there exists a modulus r such that the combinatorial sum has a nontrivia...