In this paper we confirm 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 modulo 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 sh...

In [3] and [5] the authors ask how many primes are of the form xY + yX, where gcd (x, y) = 1 and x, y 2: 2. Moreover, Jose Castillo (see [2]) asks how many primes are of the Smarandache form xil + X2 X3 + ... + Xn Xl , where n > 1, Xl, X2, ••• , Xn > 1 and gcd (Xl, X2, ••• , X n ) = 1 (see [9]). In this article we announce a lower bound for the size of the largest prime divisor of an expression...

For χ a Dirichlet character (mod q), the moments of L(s, χ) have many applications, for example to the distribution of primes in the arithmetic progressions to modulus q. The asymptotic formula of the fourth power moment in the q-aspect has been obtained by Heath-Brown [1], for q prime, and more recently by Soundararajan [5] for general q. Following Soundararajan’s work, Young [7] pushed the re...

In this paper we confirm 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 modulo 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 sh...

We analyze all the cases and propose the corresponding explicit formulae for computing 2D1 + D2 in one step from given divisor classes D1 and D2 on genus 2 hyperelliptic curves defined over prime fields. Compared with the naive method, the improved formula can save two field multiplications and three field squarings each time when the arithmetic is performed in the most frequent case. Furthermo...

According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve Γ, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of Γ. It is proven that, for a generic X , the Abel-Jacobi map of the family of elliptic sextics...

The integer d is called an exponential divisor of n = ∏r i=1 p ai i > 1 if d = ∏r i=1 p ci i , where ci|ai for every 1 ≤ i ≤ r. The integers n = ∏r i=1 p ai i ,m = ∏r i=1 p bi i > 1 having the same prime factors are called exponentially coprime if (ai, bi) = 1 for every 1 ≤ i ≤ r. In this paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors a...

Abstract: Let N be a (large) positive integer, and let b < N be a positive integer relatively prime to N whose order modulo N is large. Let QC be a quantum computer whose input register has the size specified in Shor’s original description of his order-finding algorithm. We show that when Shor’s algorithm is implemented on QC, then the probability of obtaining a divisor of the order of b modulo...

Let an arbitrarily small positive constant δ less than 1 and a polynomial f with integer coefficients be fixed. We prove unconditionally that f modulo p can be completely factored in deterministic polynomial time if p− 1 has a (ln p)O(1)-smooth divisor exceeding pδ. We also address the issue of factoring f modulo p over finite extensions of the prime field Fp and show that p− 1 can be replaced ...

In 1973, Erdős proved that a positive proportion of numbers are untouchable; that is, not of the form s(n), where s(n) := σ(n)−n is the sum of the proper divisors of n. We investigate the analogous question where σ is replaced with similar divisor functions, such as the unitary sum-of-divisors function σ∗ (which sums those divisors d of n co-prime to n/d). We use the slightly modified version o...