In working with the notion of congruence modulo $latex m$ where $latex m$ is a positive integer, one important calculation is finding the powers of a number $latex a$, i.e, the calculation $latex a…| Exploring Number Theory
Consider the quadratic congruence equation ….(1)…. where is an odd prime and is a positive integer relatively prime to . If this equation has solutions, then we say that is a quadratic residue modulo . If this equation has no … Continue reading →| Exploring Number Theory
We discuss two ways to check if a number is a Carmichael number. One way is to apply Korselt’s criterion if the prime factors of the Carmichael number are known. If the prime factors of the C…| Exploring Number Theory
In this post we discuss a beautiful connection between Fermat’s little theorem and the solvability of a quadratic congruence equation. The discussion leads to a theorem that is commonly calle…| Exploring Number Theory
In some cryptography applications such as RSA algorithm, it is necessary to compute $latex \displaystyle a^w$ modulo $latex m$ where the power $latex w$ and the modulus $latex m$ are very large num…| Exploring Number Theory
The sum of the reciprocals of the prime numbers diverges, i.e., the sum , where ranges over all the primes, diverges. Facts about prime numbers are always interesting, especially a fundamental fact such as this one. With the divergence of … Continue reading →| Exploring Number Theory
We discuss a sequence of prime numbers called the Euclid-Mullin sequence. There are actually two sequences that are called Euclid-Mullin. We focus on the first sequence but also touch on the second sequence. The first sequence is the sequence A000945 … Continue reading →| Exploring Number Theory
Prime Curios (see here) is a site that has a large collection of curiosities, wonders and trivia related to prime numbers. We would like to talk about one such curious prime number. Consider the 14…| Exploring Number Theory
The preceding two posts derive several supplements to the law of quadratic reciprocity (links are given below). We gather all the derived information in one post so that we have everything in one p…| Exploring Number Theory
The law of quadratic reciprocity shows how to flip the Legendre symbol to or along with two supplements showing how to evaluate the Legendre symbols and . The previous post gives three more supplements showing how to evaluate , and … Continue reading →| Exploring Number Theory
This is a small effort to extend the law of quadratic reciprocity. The Legendre Symbol Let be an odd prime. Let be an integer that is relatively prime to . Here’s the definition of the Legendre symbol. The bottom value … Continue reading →| Exploring Number Theory
If the modulus is small and if primitive roots modulo exist, finding primitive roots can be a simple matter. In such cases, we can simply try out all possible candidate primitive roots. For large , there is no easy or … Continue reading →| Exploring Number Theory
We show how to find square roots modulo an odd composite number by building upon the tools and techniques from three previous posts. The square roots obtained here are put together using the Chinese Remainder Theorem (CRT) based on the … Continue reading →| Exploring Number Theory