The hottest Number Theory Substack posts right now

Repunits are numbers made only of the digit 1 and can appear in different bases. For example, the number 31 can be written as 111 in base 5 and base 2.
Mersenne primes are special numbers of the form 2^p - 1 that can be prime, where p is also a prime. However, it's rare for these to actually be prime numbers.
One interesting link is between Mersenne primes and perfect numbers, which are those that equal the sum of their divisors. Each Mersenne prime corresponds to a perfect number, like how 31 corresponds to the perfect number 496.

Imaginary numbers like $i$ are not real because there isn't a real number whose square is -1, pushing the boundaries of mathematical concepts beyond reality.
The rule stating that multiplying two negative numbers results in a positive number, like $(-1) * (-1) = +1$, is a construct by mathematicians to maintain consistency in arithmetic.
Mathematicians create and manipulate rules in pure math to explore interesting results and sometimes stumble upon practical applications, demonstrating the power of abstraction in mathematics.

26 is a unique natural number sandwiched between a square and a cube, a discovery by Fermat, a notable French mathematician.
Euler's proof on Fermat's equation $y^2 = x^3 - 2$ showcases the power of abstract algebra and group theory in solving complex mathematical problems.
Understanding algebraic structures like groups, rings, and unique factorization plays a crucial role in various fields, from cryptography to machine learning.

Check for divisibility between two numbers by comparing their multiples with another common multiple, often using integers.
Composite integers have prime divisors that are less than or equal to the square root of the composite integer.
The number of multiples of a given integer within a specified interval can be calculated using floor functions and division.

The book "A computational introduction to number theory and algebra" is recommended as an excellent resource for those interested in number theory, algebra, and cryptography, particularly from a computer science perspective.
The book emphasizes computational aspects, presents algorithms, and discusses complexity analysis, making it a valuable resource for cryptography applications.
The author has created a solutions manual for some chapters of the book, focusing on exercises related to basic properties of integers, congruences, and computing with large integers.

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