The Markoff conjecture states that given a positive integer c, there is at most one triple (a, b, c) of positive integers with a ≤ b ≤ c that satisfies the equation a2 + b2 + c2 = 3abc. The conjecture is known to be true when c is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant d = 9c2 − 4,...

The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the pro...

In this paper we prove a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p = 2m + 1 is a prime congruent to 3 mod 4 if and only if Tm = m(m + 1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p = x + 8(y + z) for no odd integers x, y, z. We also sho...

The divisor function σ(n) denotes the sum of divisors positive integer n. For a prime p and m∈N, p-adic valuation m is highest power which divides m. Formulas for νp(σ(n)) are established. p=2, these involve only odd primes dividing These expressions used to establish bound ν2(σ(n))≤⌈log2⁡(n)⌉, with equality if n product distinct Mersenne primes, an p, νp(σ(n))≤⌈logp⁡(n)⌉, related solutions Lju...

The aim of this paper is to show that the computation of the discrete logarithm in the m-torsion part of the divisor class group of a curve X over a finite field ko (with char(/co) prime to m), or over a local field k with residue field /co , can be reduced to the computation of the discrete logarithm in /co(Cm)* • For this purpose we use a variant of the (tame) Täte pairing for Abelian varieti...

Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \operatorname{lcm} \{f(n)\mid n<x\} \sim (d-1)x\log x.$ In this article, we investigate the analogue prime arguments, namely, $\operatorname{lcm} \{f(p)\mid p<x\}$ where $p$ denotes a and obtain non-trivial lower bounds on it. Further, also show some results regarding greatest divisor ...

as N → ∞, where γ is the Euler-Mascheroni constant. Let us introduce a more refined notion of divisibility. A divisor k of n is unitary if k and n/k are coprime, that is, if gcd(k, n/k) = 1. This condition is often written as k||n. The number d∗(n) of unitary divisors of n is 2, where ω(n) is the number of distinct prime factors of n. This fact is easily seen to be true: If p1 1 p a2 2 · · · pr...

where p is a prime, is any fixed positive number, and (m,n,p) denotes the greatest common divisor of m, n, and p. But for an arbitrary composite number q, we do not know how large |S(m,n,χ,q)| is. In fact the value of |S(m,n,χ,q)| is quite irregular if q is not a prime. The main purpose of this paper is to obtain some good distribution properties of |S(m,n,χ,q)| in some weight mean value proble...

We sharpen a 1980 theorem of Erdős and Wagstaff on the distribution of positive integers having a large shifted prime divisor. Specifically, we obtain precise estimates for the quantity N(x, y) := #{n ≤ x : `− 1 | n for some `− 1 > y, ` prime}, in essentially the full range of x and y. We then present an application to a problem in arithmetic statistics. Let TCM(d) denote the largest order of a...

In view of the problems that the plaintext space is too small in the existing schemes. In this paper, a new improved scheme is presented by improving the DGHV scheme. The plaintext space of the improved scheme is extended from finite prime field F2 in the original scheme to finite prime field Fp. Combine and apply the method of encryption in the batch encryption scheme was proposed in 2013, and...