# 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.

"May all beings be happy and peaceful. Thank you, Lord GOD! Praise you, Lord GOD! Love you, Lord GOD! Trust you, Lord GOD! Bless you, Lord GOD! Amen!" 😂 P.S. Go Blue! 👍 ✌

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

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