The irrationality of the Euler-Mascheroni constant (𝛾) has remained a significant unresolved problem in number theory for over 250 years. This paper presents a constructive proof of the irrationality of 𝛾 by leveraging Sondow’s infinite series representation and the associated trap criteria for rational approximation. We establish that the existence of a rational representation 𝛾 = p/q necessitates the existence of an integer Z such that 0 < Z < 1, a fundamental logical impossibility. By encoding the series growth and tail estimates as machine-checked proofs, we eliminate the historical risk of errors in analytic remainder bounds. The resulting proof provides a computationally certain verification that 𝛾 is irrational.
Jonathan ƒ(n) Reed (Thu,) studied this question.