# Is the Euler–Mascheroni constant an algebraic number?

### Is the Euler–Mascheroni constant an algebraic number?

Is the Euler–Mascheroni constant ($$\gamma$$) an algebraic number?

Hmm. Is 1. $$\gamma = \frac{\sqrt{z^{2} - 1}}{2}$$?

Remark: We tentatively assume $$z$$ is an algebraic number where $$1.5 < z < 1.6$$.

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

Warning: This a difficult problem! We suspect there's much more to the true nature of the beast ($$z$$)... It may have deep secrets that are not easily discovered.

Dave.

Relevant Reference Links:

https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant;

https://en.wikipedia.org/wiki/Newton%27s_method.
Attachments Is the Euler–Mascheroni constant an algebraic number?
Euler–Mascheroni constant.jpg (52.04 KiB) Viewed 303 times
Guest

### Re: Is the Euler–Mascheroni constant an algebraic number?

FYI: "Algebraic and Transcendental Numbers" by Prof. S. Fischler,

https://www.imo.universite-paris-saclay.fr/~fischler/inde/inde_fischlerpondichery.pdf.
Guest

### Re: Is the Euler–Mascheroni constant an algebraic number?

Guest wrote:Is the Euler–Mascheroni constant ($$\gamma$$) an algebraic number?

Hmm. Is 1. $$\gamma = \frac{\sqrt{z^{2} - 1}}{2}$$?

Remark: We tentatively assume $$z$$ is an algebraic number where $$1.5 < z < 1.6$$.

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

Warning: This is a difficult problem! We suspect there's much more to the true nature of the beast ($$z$$)... It may have deep secrets that are not easily discovered.

Dave.

Relevant Reference Links:

...

https://en.wikipedia.org/wiki/Newton%27s_method.

Important Reference Link:

"EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS" by Professor J. C. Lagarias is an excellent/fruitful resource for answering our question (Is the Euler–Mascheroni constant ($$\gamma$$) an algebraic number or a transcendental number?).

Source Link: https://arxiv.org/pdf/1303.1856.pdf.

Now we feel can confident we can crack our problem soon.

GOOD LUCK! Go Blue! Guest

### Re: Is the Euler–Mascheroni constant an algebraic number?

Oops! ... We can feel confident we can crack our problem soon...
Guest

### Re: Is the Euler–Mascheroni constant an algebraic number?

How soon?

Hmm. We can crack this problem before April 15, 2021 (The 314th Birthday of the great one, Leonhard Euler).

Hmm. The number, 314, must be a lucky number for a number of reasons... Guest

### Re: Is the Euler–Mascheroni constant an algebraic number?

Valid Assumption (Convergent): 1. $$\gamma = \frac{\sqrt{z^{2} - 1}}{2}$$ where $$z$$ is the hypotenuse of a right triangle and $$x = 1$$ and $$y = 2 \gamma$$ are the legs of our right triangle.
Attachments The area of the blue region converges to the Euler-Mascheroni constant....png (3.69 KiB) Viewed 248 times
Guest

### Re: Is the Euler–Mascheroni constant an algebraic number?

Guest wrote:Valid Assumption (Convergent): 1. $$\gamma = \frac{\sqrt{z^{2} - 1}}{2}$$ where $$z$$ is the hypotenuse of a right triangle and $$x = 1$$ and $$y = 2 \gamma$$ are the legs of our right triangle.

We assume that $$\gamma$$ and $$z$$ are algebraic (rational) numbers.
Guest

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