[Math] Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental

nested-radicalstranscendental-numbers

I thought it was easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational, but found a gap in my proof. Simple finite approximations show the denominator cannot be small, though, strongly suggesting irrationality. However, can it be shown whether this number is algebraic or transcendental? My hunch is that it's transcendental but I have no idea on how to start such a proof (especially since I can't fill my gap to prove irrationality).

Best Answer

This is not an answer, just a list of papers concerned with this constant; it's too long for a comment. Going by the reviews of the papers, they deal with efficient methods for calculating the constant, and not (directly) with questions of irrationality and/or transcendence. The first paper on the list is too new to have been reviewed at this point.

MR3349435
Lu, Dawei; Song, Zexi; Some new continued fraction estimates of the Somos' quadratic recurrence constant. J. Number Theory 155 (2015), 36–45.

MR3019753
Chen, Chao-Ping New asymptotic expansions related to Somos' quadratic recurrence constant.
C. R. Math. Acad. Sci. Paris 351 (2013), no. 1-2, 9–12.

MR2825112 (2012h:11181)
Hirschhorn, Michael D. A note on Somos' quadratic recurrence constant.
J. Number Theory 131 (2011), no. 11, 2061–2063.

MR2809034 (2012e:05038)
Nemes, Gergő On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant.
Appl. Anal. Discrete Math. 5 (2011), no. 1, 60–66.

MR2684487 (2011i:11179)
Mortici, Cristinel Estimating the Somos' quadratic recurrence constant.
J. Number Theory 130 (2010), no. 12, 2650–2657.

MR2319662 (2008f:40013)
Sondow, Jonathan; Hadjicostas, Petros The generalized-Euler-constant function $\gamma(z)$ and a generalization of Somos's quadratic recurrence constant.
J. Math. Anal. Appl. 332 (2007), no. 1, 292–314.

MR2262724 (2008b:11081)
Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana Arithmetical properties of some series with logarithmic coefficients.
Math. Z. 255 (2007), no. 1, 117–131.

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