The difference between distinct harmonic numbers is never an integer.

The problem was to characterize all pairs of harmonic numbers differing by 1.

With possible selection of harmonic numbers to be filtered or not (example to keep fundamental).

Harmonic numbers were studied in antiquity and are important in various branches of number theory.

Using the asymptotics of the harmonic numbers, we obtain:

In the following, and denote harmonic numbers.

Some derivatives of fractional harmonic numbers are given by:

In number theory, the harmonic numbers are the sums of the inverses of integers, forming the harmonic series.

As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers:

These numbers are named after Joseph Wolstenholme, who proved Wolstenholme's theorem on modular relations of the generalized harmonic numbers.

The sum in this equation involves the harmonic numbers, H. Expanding some of the terms in the Hurwitz zeta function gives:

The related sum occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of Stirling numbers.

All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.

The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős-Graham conjecture and the Erdős-Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.

One year later, at the request of the bishop of Meaux, he wrote The Harmony of Numbers in which he considers a problem of Philippe de Vitry involving so-called harmonic numbers, which have the form 2 3.

The distance travelled on the last trip is the nth harmonic number, H. As the harmonic numbers are unbounded, it is possible to exceed any given distance on the final trip, as along as sufficient fuel is available at the base.