Almost all hyperharmonic numbers are not integers

It is an open question asked by Mezö that there is no hyperharmonic integer except 1. So far it has been proved that all hyperharmonic numbers are not integers up to order r=25. In this paper, we extend the current results for large orders. Our method will be based on three different approaches, namely analytic, combinatorial and algebraic. From analytic point of view, by exploiting primes in short intervals we prove that almost all hyperharmonic numbers are not integers. Then using combinatorial techniques, we show that if n is even or a prime power, or r is odd then the corresponding hyperharmonic number is not integer. Finally as algebraic methods, we relate the integerness property of hyperharmonic numbers with solutions of some polynomials in finite fields.