Methods for finding factors of large integers

Methods of Factoring Large Integers

Integers Free of Large Prime Factors

Deene (x;y) to be the number of positive integers n x such that n has no prime divisor larger than y. We present a simple algorithm that approximates (x; y) in O(yf log log x log y + 1 log log y g) oating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other kno...

Finding Factors of Factor Rings over the Gaussian Integers

1. INTRODUCTION AND HISTORY. The Gaussian integers are defined to be the set Z[i] = {a + bi : a, b ∈ Z, i = √ −1}. These sit inside the complex numbers C and thus obey the usual rules of addition and multiplication; indeed, despite the presence of the imaginary i, they are quite similar to the " traditional " integers. In fact, in the set Z[i] one can define (Gaussian integer) primes, construct...

Approximating the number of integers free of large prime factors

Define Ψ(x, y) to be the number of positive integers n ≤ x such that n has no prime divisor larger than y. We present a simple algorithm that approximates Ψ(x, y) in O(y{ log log x log y + 1 log log y }) floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than o...

islanding detection methods for microgrids

امروزه استفاده از منابع انرژی پراکنده کاربرد وسیعی یافته است . اگر چه این منابع بسیاری از مشکلات شبکه را حل می کنند اما زیاد شدن آنها مسائل فراوانی برای سیستم قدرت به همراه دارد . استفاده از میکروشبکه راه حلی است که علاوه بر استفاده از مزایای منابع انرژی پراکنده برخی از مشکلات ایجاد شده توسط آنها را نیز منتفی می کند . همچنین میکروشبکه ها کیفیت برق و قابلیت اطمینان تامین انرژی مشترکان را افزایش ...