The aim of this article is to study the system piecewise linear difference equations xn+1=|xn|−yn−b and yn+1=xn−|yn|+1 where n≥0. A global behavior for b=4 shows that all solutions become equilibrium point. For a large value |x0| |y0|, we can prove (i) if b=5, then solution becomes point (ii) b≥6, periodic prime period 5.

We prove that if q is a power of a prime p and p divides a, with k ≥ 0, then 1 + (q − 1) ∑ 0≤b(q−1)<a ( a b(q − 1) ) ≡ 0 (mod p). The special case of this congruence where q = p was proved by Carlitz in 1953 by means of rather deep properties of the Bernoulli numbers. A more direct approach produces our generalization and several related results.

Let φ(z) ∈ Q(z) be a rational function of degree d ≥ 2 with φ(0) = 0 and such that φ does not vanish to order d at 0. Let α ∈ Q have infinite orbit under iteration of φ and write φ(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, w...

Dedicated to Benjamin Fine on the occasion of his 60th birthday. The object of this paper is to give a proof of the following theorem: S/P ∼ = Λ mn ⊆ Q + , where S/P is a certain torsion-free factor group of the Baumslag-Solitar group a, b; a −1 b m a = b n | m = 0, n = 0, m, n ∈ Z, with m and n are relatively prime, and Λ mn is a subgroup of the additive group of the rational numbers Q + .

Abstract The TiO B 3 Π − X Δ electronic transition ( γ ′ system) is an important opacity source in the atmospheres of M dwarfs and hot Jupiter exoplanets. 0–0, 1–0, 2–1 bands band s...

Fix a prime p. For nonnegative integers a, b, and d, we seek a formula for the number of subgroups of order pd in Z/paZ× Z/pbZ. Set Na,b,d = #{H ⊂ Z/pZ× Z/pZ : #H = p}. This is symmetric in a and b (Na,b,d = Nb,a,d), so when it is convenient we can limit attention to the case a ≤ b. Trivially Na,b,d = 0 if d > a + b, so we may assume 0 ≤ d ≤ a + b. For 1 ≤ a ≤ b, and a + b ≥ d, we will see that...

Motivated by the first search for rare charged-current $B$ decay to four leptons, $\ell \bar{\nu}_\ell \ell^{(\prime)} \bar{\ell}^{(\prime)}$, we calculate amplitude with factorization methods. We obtain $B\to \gamma^*$ form factors, which depend on invariant masses of two lepton pairs, at leading power in an expansion $\Lambda_{\rm QCD}/m_b$ next-to-leading order $\alpha_s$, and $\mathcal{O}(\...

b + an−1b + · · ·+ a0 = 0. Example 1.2. Let A = Z and B = Q. Then b ∈ Q is integral over Z if and only if b ∈ Z. Indeed, if b ∈ Z then b solves x − b ∈ Z[x]. Conversely, write b = c/d for relatively prime integers c and d, d > 0. Let f(x) = x + an−1x + · · ·+ a0 be a polynomial with integer coefficients that b satisfies. Substituting b for x and multiplying by d we obtain c = −(dan−1c + · · ·+ ...

When numbers are added in base b in the usual way, carries occur. If two random, independent 1-digit numbers are added, then the probability of a carry is b−1 2b . Other choices of digits lead to less carries. In particular, if for odd b we use the digits {−(b− 1)/2,−(b− 3)/2, . . . , . . . (b− 1)/2} then the probability of carry is only b −1 4b2 . Diaconis, Shao and Soundararajan conjectured t...