On Quadratic Polynomials Rich in Prime Numbers

Prime numbers and methods of their generation have attracted mathematicians for centuries and, in the digital age, have found their applications in cryptography, signal processing and data compression, secure communications, hashing algorithms, cybersecurity, quantum computing algorithms, blockchain technology, and other areas. Prime numbers and prime generating polynomials were studied in [1-20]. There are many prime generating polynomials of different degrees \cite{weisstein2005} found so far; the most famous of them is $x^2+x+41$ found by Euler in 1772 and $x^2-x+41$ found by Legendre in 1798. Researchers are curious to find such a polynomial $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ that produces more prime numbers for given integer values of $x$. This work is no exception, but the reader will not find any mathematical formulas or theorems as in mathematical works; instead, we want to show the result of our computational experiments in the programming language Julia, which in particular led to the discovery of quadratic polynomials that, similarly to Euler's prime generating polynomial, generate 40 primes. We also show that some of the currently known polynomials are not the richest in terms of the percentage of primes appearing in larger intervals, i.e. they produce fewer primes. For a better and more systematic understanding of what happens in prime number research, we demonstrate a video with network visualisation of keyword co-occurrence and co-authorship based on data from 7548 documents indexed in the Scopus database. Readers of this work are welcome to send me comments, suggestions, or proposals for collaborative research on prime numbers and their applications in science and engineering.