Small Prime Factors Research Articles

On divisors of sums of integers. III

1. Let P{ή) and p(n) denote the greatest and smallest prime factor of n, respectively. Recently in several papers, Balog, Erdόs, Maier, Sarkozy, and Stewart have studied problems of the following type: if A\,...,Ak are dense sets of positive integers, then what can be said about the arithmetical properties of the sums H h % with a, G A\,...,ak G Akl In particular, Balog and Sarkόzy proved that there is a sum a\+ci2 (#i G A\9 aι G A2) for which P(a\+aι) is small, i.e., all the prime factors of +aι are small. On the other hand, Balog and Sarkόzy and Sarkόzy and Stewart studied the existence of a sum CL Λ h % for which P(a H h ak) is large. In this paper we study p(a H \ak). Our goal is to show that if A\,..., Ak are sets of positive integers then there exists a sum + h cik with G A\,..., ak € Ak that is divisible by a prime. In the most interesting special case, namely A = = Ak, there are sums CL Λ Vak divisible by /c, so that p(a H h ak) 2) by studying the case min, \At > N /+. Here Gallagher's larger sieve will be used. The results in §§3 and 4 do not give especially good results when the sets A\,...,Ak are very dense. In §5, we will give an essentially best possible result for the small prime factors of the sums d Λ h ak in the case when (\AX \Ak\) / > Nexp(-clogklogN/loglogN)

Computing Powers in Parallel

Fast parallel computations are presented for large powers modulo an element that has only small prime factors. They work for integers and polynomials over small finite fields.

Refined analysis and improvements on some factoring algorithms

By combining the principles of known factoring algorithms we obtain some improved algorithms which by heuristic arguments all have a time bound O( exp c ln n ln lnn ) for various constants c ⩾3. In particular, Miller's method of solving index equations and Shanks' method of computing ambiguous quadratic forms with discriminant − n can be modified in this way. We show how to speed up the factorization of n by using preprocessed lists of those numbers in [− u, u] and [ n - u, n + u], 0 ⪡ u ⪡ n which only have small prime factors. These lists can be uniformly used for the factorization of all numbers in [ n - u, n + u]. Given these lists, factorization takes O( exp[2(ln n) 1 3 ( ln ln n) 2 3 ] steps. We slightly improve Dixon's rigorous analysis of his Monte Carlo factoring algorithm. We prove that this algorithm with probability sol1 2 detects a proper factor of every composite n within o( exp 6 ln n ln lnn ) steps.

Gaussian integers with small prime factors

Let ψG(xt, x) denote the number of Gaussian integers with norm not exceeding x2t whose Gaussian prime factors have norm not exceeding x2. Previous estimates have required restrictions on the parameter t with respect to x. The purpose of this note is to present asymptotic estimates for ψG(xt, x) for all ranges of the parameter t with respect to x.

An improved algorithm for computing logarithms over<tex>GF(p)</tex>and its cryptographic significance (Corresp.)

A cryptographic system is described which is secure if and only if computing logarithms over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(p)</tex> is infeasible. Previously published algorithms for computing this function require <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(p^{1/2})</tex> complexity in both time and space. An improved algorithm is derived which requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O =(\log^{2} p)</tex> complexity if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p - 1</tex> has only small prime factors. Such values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> must be avoided in the cryptosystem. Constructive uses for the new algorithm are also described.

On ideals having only small prime factors

On ideals having only small prime factors

Duality between prime factors and an application to the prime number theorem for arithmetic progressions

We study in this paper a new duality identity between large and small prime factors of integers and its relationship with the prime number theorem for arithmetic progressions. The asymptotic behavior of large prime factors of integers leads to interesting relations involving the Möbius function.

On k-Free Integers with Small Prime Factors

On k-Free Integers with Small Prime Factors

On 𝑘-free integers with small prime factors

The object of this note is to give a nontrivial lower estimate for the function ψ k ( x , y ; h ) {\psi _k}(x,y;h) , defined to be the number of k k -free integers m m such that 1 ⩽ m &gt; x , ( m , h ) = 1 1 \leqslant m &gt; x,\,(m,h) = 1 , and m m has no prime factor greater than or equal to y y .

Numbers with small prime factors, and the least 𝑘th power non-residue

Numbers with small prime factors, and the least 𝑘th power non-residue