Some number theory problems related to binomial coefficients

In this talk, we first introduce congruences of E. Lehmer, Morley, Jacobstahl related to binomial coefficients, and generalize them from those modulo prime powers to modulo integer powers. As an application, we will show their links to the three types of complete elliptic integrals. Then I will use binomial coefficients in the representation of prime numbers by binary forms. Finally, we define figurate primes by binomial coefficients, and use them in the variety of Hilbert’s 8thproblem.