Rationality of the Euler–Mascheroni constant

Determine whether the Euler–Mascheroni constant γ, defined as the limit of the sequence b_n = ∑_{m=1}^{n} 1/m − log(n), is rational or irrational.

Background

In Section 3.2, while discussing examples of task types related to sequences and Cauchy convergence, the authors note that the sequence b_n = a_n − log(n), where a_n = 1 + 1/1 + 1/2 + … + 1/n, is Cauchy and hence convergent.

They then remark in a footnote that the limit of this sequence is the Euler–Mascheroni constant γ and explicitly state that it is still unknown whether γ is rational or irrational.

References

The sequence (b_n)n with b_n=a_n-\log{\left(n\right)}=\sum{m=1}{n}\frac{1}{m}-\log{\left(n\right)} is a Cauchy sequence (so convergent). Curiously enough, it is still not known whether its limit γ (the so-called Euler-Mascheroni constant) is rational or irrational.

— Didactic analysis of the modality of study of the real numbers in the Degree in Mathematics (2509.20023 - Espín et al., 24 Sep 2025) in Section 3.2 (Current didactic ends, means and phenomena), footnote