Let O be a maximal order in a definite quaternion algebra over Q of prime discriminant p, and ` a small prime. We describe a probabilistic algorithm, which for a given left O-ideal, computes a representative in its left ideal class of `-power norm. In practice the algorithm is efficient, and subject to heuristics on expected distributions of primes, runs in expected polynomial time. This breaks...

let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

We present a new algorithm that nds all primes up to n using at most O(n= log log n) arithmetic operations and O(n=(log n log log n)) space. This algorithm is an improvement of a linear prime number sieve due to Pritchard. Our new algorithm matches the running time of the best previous prime number sieve, but uses less space by a factor of (log n). In addition, we present the results of our imp...

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a $A$ universally bounded all choices $A$. In 1988 he asked this universal bound attained by prime numbers. One source difficulty conjecture $\sum n^{-\lambda}$ maximized primes and only $\lambda$ at least critical exponent...

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of n. In particular, we show that P(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and P(τ(p)τ(p2)τ(p3)) > (1+o(1)) log log p log log log p loglog log log p for every prime p with τ(p) 6= 0.

A prefix-free language is a prime if it cannot be decomposed into a concatenation of two prefix-free languages. We show that we can check in polynomial time if a language generated by a simple contextfree grammar is a prime. Our algorithm computes a canonical representation of a simple language, converting its arbitrary simple grammar into Prime Normal Form (PNF); a simple grammar is in PNF if ...

To be more precise, let F be a totally real field of degree g, of narrow class number 1, and B/F a quaternion algebra split at exactly one infinite place ∞1 of F . Let N be an integral ideal prime to the discriminant of B. Let us write Γ, Γ0(N ), Γ1(N ), Γ(N ) for the corresponding congruence subgroups of Γ, the image in PGL2(R) of the positive norm units of a maximal order O (unique up to conj...