Is Euler-Mascheroni Constant Irrational?

Open Question

It is not even known whether the Euler-Mascheroni constant $\gamma$ is irrational, let alone whether it is transcendental or not.


Progress

In $1977$ it was established by Richard Peirce Brent that if $\gamma$ is a rational number expressible as the ratio of two integers $\dfrac a b$, it would be necessary for $b$ to be greater than $10^{10 \, 000}$.

By $1980$ that lower limit had been raised, according to Brent in collaboration with Edwin M. McMillan, to $10^{15 \, 000}$.


Sources

  • 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
  • 1977: Richard P. Brent: Computation of the Regular Continued Fraction for Euler's Constant (Math. Comp. Vol. 31: pp. 771 – 777)  www.jstor.org/stable/2006010
  • 1980: Richard P. Brent and Edwin M. McMillan: Some New Algorithms for High-Precision Computation of Euler's Constant (Math. Comp. Vol. 34: pp. 305 – 312)  www.jstor.org/stable/2006237
  • 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,57721 56649 \ldots$
  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 577 \, 215 \, 664 \, 901 \, 532 \, 860 \, 606 \, 512 \, 090 \, 082 \, 402 \, 431 \ldots$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 57721 \, 56649 \, 01532 \, 86060 \, 65120 \, 90082 \, 40243 \, 1 \ldots$
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