Sequences and special functions in number theory and combinatorics

Much of the proposed research can be described as "classical", dealing with mathematical objects that have interested mathematicians for decades, in some cases even centuries. However, new methods make it worthwhile to take a fresh look at old objects, with the hope of obtaining new results. Also, advances in computer technology and in algorithms make it possible to do "mathematical experiments", sometimes at a large scale, and use the outcomes as a basis for theoretical investigations which often lead to new and sometimes unexpected results.The objects to be studied in this proposal include prime numbers and congruences, factors of special integers, polynomials, infinite series, and various special sequences of numbers and functions. The expected results contribute to a deeper understanding of the objects under investigation and have no immediate applications outside of mathematics. However, as is usually the case when one deals with large integers, prime numbers, or polynomials, there is always the possibility of applications in cryptography, the science behind data security. Although this is not the main purpose of the proposed research, I will keep such possible applications in mind.