# Is the Euler–Mascheroni constant an algebraic number or a transcendental number?

Valid Assumption (Convergence): $1. \gamma = \frac{\sqrt{z^{2} - 1}}{2}$.

Remark: We tentatively assume $z$ is an algebraic number (rational number) where $1.52 < z < 1.53$.

We can apply Newton’s Method to equation one and study its convergence with regards to $\gamma$.

Warning: This may be a difficult problem…

If $x = \gamma$ is algebraic, then $\sum_{i=0}^{j } a_{i } x^{i } = 0$ where $a_{k } \in \mathbb Q$ with $j > 0$. Right?

1. Does $\gamma = | \frac{ a_{0 } }{a_{1 }}|$?
2. Does $\gamma =\frac{ -a_{1 } \pm \sqrt{a_{1 } ^2 -4a_{2 }a_{0 } } }{2a_{2 }}$?

This may be a good start. We should investigate further. Good Luck!

Dave.

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## 6 thoughts on “Is the Euler–Mascheroni constant an algebraic number or a transcendental number?”

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