The purpose of these notes is to illustrate the descent technique used in [Maz72] and [Maz77] to bound the rank of abelian varieties A/Q; this method is used in [Maz77] to prove that the Eisenstein quotient of J0(N)/Q (where N is a prime number) has rank 0. We begin with a brief review of the standard method of descent. We then explain how fppf cohomology can be used following this model to pro...

For α, β, γ, δ ∈ Z and ν = (α, β, γ, δ), the q-Fibonacci numbers are given by F ν 0 (q) = 0, F ν 1 (q) = 1 and F ν n+1(q) = q αn−βF ν n (q) + q γn−δF ν n−1(q) for n > 1. And define the q-Lucas number Ln(q) = F ν n+1(q)+ q γ−δF ν∗ n−1(q), where ν∗ = (α, β− α, γ, δ − γ). Suppose that α = 0 and γ is prime to n, or α = γ is prime to n. We prove that Ln(q) ≡ (−1) (mod Φn(q)) for n > 3, where Φn(q) i...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prim...

Let p > 2 be a prime number. For each integer 0 < n < p, define n by the congruence nn ≡ 1 (mod p) with 0 < n < p. We are led to study the distribution behavior of n− n+ 1 in order to prove the asymptotic formula p−2 ∑ n=1 |n− n+ 1| = 1 3 p +O(p log p).

Suppose a 1 < a 2 < : : : < a Z are distinct integers in f1; : : :; N g. If a 0 +" 1 a 1 +" 2 a 2 + + " Z a Z is prime for all choices of " 1 ; " 2 ; : : : ; " Z 2 f0; 1g, then Z (9=2 + o(1)) log N= log log N. The argument uses a modiied form of Gallagher's Larger Sieve, and shows that if Z > (9=2 + o(1)) log N= log log N , then for some prime p < Z 2 , the subset sums " 1 a 1 + " 2 a 2 + + " Z...

This work studies the existence of positive prime periodic solutions of higher order for rational recursive equations of the form yn = A + yn−1 yn−m , n = 0, 1, 2, . . . , with y−m , y−m+1, . . . , y−1 ∈ (0,∞) and m ∈ {2, 3, 4, . . .}. In particular, we show that for sufficiently small A > 0, there exist periodic solutions with prime period 2m +Um + 1, for almost all m, where Um = max{i ∈ N : i...

The sequence of (terminating balanced) hypergeometric sums an = n ∑ k=0 ( n k )2( n+ k k )2 , n = 0, 1, . . . , appears in Apéry’s proof of the irrationality of ζ(3). Another example of hypergeometric use in irrationality problems is Ramanujan-type identities for 1/π, like ∞ ∑ k=0 ( 2k k )3 (4k + 1) (−1) 26k = 2 π . These two, seemingly unrelated but both beautiful enough, hypergeometric series...

As usual we denote the number of integers not exceeding n and relatively prime to n by Euler's 0 function 0(n). Lehmer 2) calls the A (n) integers 1=a,< a 2 <. .. k. Lehmer) says that the totatives are uniformly distributed with respect to k. To shorten the notation we say that T(n, k)...

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1...

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, ...,+∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characteri...