Over time I encountered a lot of interesting mathematical open problems that caught my attention. At some level I tried to understand and/or solve (in vain) them, but they remained in my mind and I hope at some point of my life to see their solution. The first two of these (Riemann Hypothesis and P… Continue reading Challenging mathematical open problems
Category: Mathematics
Computational Models
Computers are very fast circuits able to do several basic logical, mathematical and memory operations and we use long sequences of these operations to perform almost every possible task. They are so fast that we think the elaboration as instantaneous, but if the input is large it may require some time. I still remember the… Continue reading Computational Models
One million digits of Pi
Let continue the serie of computation of mathematical constants that we started with Euler Number. This time our target is π, the famous “3.14” ratio between circumference and diameter of a circle. Since out target is to compute one million digits in few minutes this post will use concept of code optimization. Like e, π… Continue reading One million digits of Pi
Mersenne Primes and Perfect Numbers
Primes are a fascinating group of integer numbers. They definition is short and clear (a prime number is a number that can be divided without remainder only by itself and 1), there are infinite primes and “quite frequent” (for number of n digits they are about n/(log n), so one number every log n is… Continue reading Mersenne Primes and Perfect Numbers
Computing 1 million digits of Euler’s Number
With this thread I start a new mathematical and computational branch: the art of computing decimals of particular mathematical constants. The first one will be e, the Euler’s number, about 2.7182818284, which is the base of exponential function ex, the only function equal to its derivative. Its inverse is the natural logarithm ln x, or… Continue reading Computing 1 million digits of Euler’s Number
Integer Factoring – Quadratic Sieve
SPOILER ALERT!!! Long and boring article, quite technical, it is in preparation to a Java implementation of the QS that will be included in the Factoring App presented in this site. So far the Java library is not yet ready, but it is under development. In previous posts we saw something about integer factorization, in… Continue reading Integer Factoring – Quadratic Sieve
The game of life, ChatGPT version
Organic life is based on chemical reactions. Given a initial state of some compound we can try to guess how it will evolve emulating the interactions of molecules (think for example to proteins unfolding). Tha game of life is a mathematical model that wants to use this principle: an initial state and single rules to… Continue reading The game of life, ChatGPT version
Pollard’s Rho Factoring Method
Finally it comes the weekend and you can go to the mountain, but as soon as you exit home and you reach the car you find a 2256+1 sleeping on the hood. You try to move it but it is too heavy. Oh no, again! you’ll have to factor it so you can move smaller… Continue reading Pollard’s Rho Factoring Method
Trial Division and Eratosthene’s Sieve
Let’s suppose you change telephone number. Almost surely the first thing you think is “is my new phone number prime?”. Since the number is small the naive method “let’s try to divide the phone number by all numbers less than it” is good (at least if you do it with the computer). You can improve… Continue reading Trial Division and Eratosthene’s Sieve
