Published inIntuition·Jan 24Fun with Numbers - Kaprekar’s ConstantLet’s have some fun with numbers. Pick a four-digit number with at least two distinct digits(monodigit numbers like 1111, 8888, are not going to work). Apply the following operations on the chosen number: Rearrange the digits in descending order - this will give the biggest number that can be formed…Mathematics4 min readMathematics4 min read
Published inIntuition·Oct 26, 2022Binary Quadratic Forms and Primes of the Form x² + ny²Here is an interesting observation about numbers: Some prime numbers can be written as sums of two squares(x² + y²), like: 5 = 1² + 2² 13 = 2² + 3² 17 = 1² + 4² 29 = 2² + 5² You’ll notice that such primes are of the form 4k + 1, where k…Mathematics13 min readMathematics13 min read
Published inIntuition·Jun 30, 2022The Idea of IdealsAt the end of the post Unique and Non-unique Factorization, we saw how the introduction of Kummer’s “ideal numbers” remedied the lack of unique factorization in a certain ring of integers. And it was pointed out that these “ideal numbers” are just that - ideal. They have no real or…Mathematics9 min readMathematics9 min read
Published inIntuition·Apr 11, 2022Unique and Non-unique FactorizationLet’s begin with an illustration. Of (non)unique factorization. Consider the following factorizations of 24: 24 = 8 x 3 24 = 12 x 2 24 = 6 x 4 Three sets of factorizations with different sets of factors in each case. So, the factorization of 24 is not unique? It can be…Mathematics7 min readMathematics7 min read
Mar 30, 2022Number Fields“God made the integers, all else is the work of man”. - Leopold Kronecker The positive integers are the natural choice for counting, that is why they are also called natural numbers, and they form the foundation on which other kinds of numbers are built. Even, a fractional quantity can…Mathematics10 min readMathematics10 min read
Published inNerd For Tech·Feb 6, 2022Here’s How Quadratic Sieve Factorization WorksFinding the product of two or more numbers is a straightforward mathematical operation, but the inverse operation of decomposing a number into its factors is not a walk in the park. This difficulty in factorization is the basis of the RSA cryptosystem. Factorization has always been of great interest, and…Mathematics14 min readMathematics14 min read
Published inNerd For Tech·Apr 28, 2021Experimental Math — Computing Units of Modular RingsA ring is an algebraic structure modelled after the integers and their arithmetic properties. Formally, a ring is defined as a structure consisting of a nonempty set, S, and two operations; addition(+) and multiplication(*) satisfying the following properties for all x, y, z in S: x + y is also…Mathematics9 min readMathematics9 min read
Jan 27, 2021Getting Down with Fermat — A Lesson in Infinite DescentPierre de Fermat, a seventeenth century French lawyer and mathematician needs no introduction, he is famous for the notoriously hard conjecture, Fermat’s Last Theorem(FLT), which took over three centuries and a half to prove after a series of failed attempts by cranks and professional mathematicians alike. “It is impossible to…Mathematics7 min readMathematics7 min read
Nov 14, 2020A first look at GroupsA group is a monoid in which every element has an inverse. Fact of life. I always chuckle when I think of that college textbook definition of a group. Innocently waggish though very correct. While that opening definition of a group is correct, it doesn’t immediately convey the essence of…Mathematics14 min readMathematics14 min read
