Is the Euler–Mascheroni constant an algebraic number?

[Guest]

Is the Euler–Mascheroni constant ([tex]\gamma[/tex]) an algebraic number?

Hmm. Is 1. [tex]\gamma = \frac{\sqrt{z^{2} - 1}}{2}[/tex]?

Remark: We tentatively assume [tex]z[/tex] is an algebraic number where [tex]1.5 < z < 1.6[/tex].

We can apply Newton's Method to equation one and study its convergence with regards to [tex]\gamma[/tex].

Warning: This a difficult problem! We suspect there's much more to the true nature of the beast ([tex]z[/tex])... 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.

[Guest]

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

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

[Guest]

Is the Euler–Mascheroni constant ([tex]\gamma[/tex]) an algebraic number?

Hmm. Is 1. [tex]\gamma = \frac{\sqrt{z^{2} - 1}}{2}[/tex]?

Remark: We tentatively assume [tex]z[/tex] is an algebraic number where [tex]1.5 < z < 1.6[/tex].

We can apply Newton's Method to equation one and study its convergence with regards to [tex]\gamma[/tex].

Warning: This is a difficult problem! We suspect there's much more to the true nature of the beast ([tex]z[/tex])... 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 ([tex]\gamma[/tex]) 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]

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

[Guest]

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]

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

[Guest]

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

We assume that [tex]\gamma[/tex] and [tex]z[/tex] are algebraic (rational) numbers.

[Guest]

Hmm. Has anyone defined the Euler–Mascheroni constant?

[Guest]

Hmm. Has anyone defined the Euler–Mascheroni constant?

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

1. Does [tex]\gamma = | \frac{ a_{0 } }{a_{1 }}|[/tex]?

2. Does [tex]\gamma =\frac{ -a_{1 } \pm \sqrt{a_{1 } ^2 -4a_{2 }a_{0 } } }{2a_{2 }}[/tex]?

...

[Guest]

Hmm. Has anyone defined the Euler–Mascheroni constant?

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

1. Does [tex]\gamma = | \frac{ a_{0 } }{a_{1 }}|[/tex]?

2. Does [tex]\gamma =\frac{ -a_{1 } \pm \sqrt{a_{1 } ^2 -4a_{2 }a_{0 } } }{2a_{2 }}[/tex]?

...

This may be a good start. But the previous posts may be doubtful? We shall investigate further. Good Luck!

[Guest]

What are the rational values, {[tex]a_{k }[/tex]} (if they exist), such that [tex]\gamma[/tex] is algebraic?

We assume [tex]k > 0[/tex].

[Guest]

What are the rational values, {[tex]a_{k }[/tex]} (if they exist), such that [tex]\gamma[/tex] is algebraic?

We assume [tex]k > 0[/tex].

Remark: {[tex]a_{k }[/tex]} = {[tex]a_{0 }[/tex], a[tex]_{1 }[/tex], [tex]a_{2 }[/tex], ..., a[tex]_{k >0 }[/tex])

Bad notation?

[Guest]

What are the rational values, {[tex]a_{k }[/tex]} (if they exist), such that [tex]\gamma[/tex] is algebraic?

We assume [tex]k > 0[/tex].

Remark: {[tex]a_{k }[/tex]} = {[tex]a_{0 }[/tex], a[tex]_{1 }[/tex], [tex]a_{2 }[/tex], ..., a[tex]_{k >0 }[/tex])

Bad notation?

A Fix: We let A = {[tex]a_{0 }[/tex], a[tex]_{1 }[/tex], [tex]a_{2 }[/tex], ..., a[tex]_{k }[/tex]} for some k > 0 ...

Good Luck!