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May 30

A proof attempt for the transcendence of ℼ

The proposition "if $a$ is rational then $a$ is algebraic" is comprehensively true,
and is equivalent to "if $a$ is inalgebraic then $a$ is irrational" (contrapositive).

My question is this:
The proofs for the transcendence of $\pi$ are of course by contradiction.
Now, do you think it is possible to prove somehow the proposition "if $\pi$ is algebraic then $\pi$ is rational", reaching a contradiction?
Meaning, by assuming $\pi$ is algebraic and using some of its properties, can we conclude that it must be algebraic of degree 1 (rational) – contradicting its irrationality?

I know the proposition "if $a$ is algebraic then $a$ is rational" is not comprehensively true ( ${\sqrt {2}}$ is a counterexample),
but I am basically asking if there exist special cases $a\in \mathbb {R}$ such that it does hold for them. יהודה שמחה ולדמן (talk) 18:48, 30 May 2024 (UTC)[reply]

There are real-valued expressions

E

such that the statement "if $E$ is algebraic, $E$ is rational" is provable, but this does not by itself establish transcendence. For example, substitute

{\tfrac {22}{7}}

for

E.

Given the irrationality of

\pi ,

proving the implication for

\pi

would give yet another proof of the transcendence of

\pi

. I see no plausible approach to proving this implication without proving transcendence on the way, but I also see no a priori reason why such a proof could not exist. --Lambiam 19:24, 30 May 2024 (UTC)[reply]

Also, for a while now I am looking to prove the transcendence of

\pi

by trying to generalize Bourbaki's/Niven's proof that π is irrational for the

n

th-degree polynomial:

{\begin{aligned}&\theta =0.5\pi \\[5pt]&p(\theta )=a_{0}+a_{1}\theta +a_{2}\theta ^{2}+\cdots +a_{n}\theta ^{n}=0\\[5pt]&f(x)={\frac {{\bigl [}\,p(x)\,p(-x)\,{\bigr ]}^{m}}{m!}}\\&I=\int \limits _{-\theta }^{\theta }f(x)\cos(x)dx=2\int \limits _{0}^{\theta }f(x)\cos(x)dx=2\sum _{k\,=\,0}^{nm}(-1)^{k}f^{(2k)}\!(\theta )\end{aligned}}

Unfortunately, I failed to show that

I

is a non-zero integer (aiming for a contradiction).

Am I even on the right track, or is my plan simply doomed to fail and I am wasting my time?

Could the general Leibniz rule help here? יהודה שמחה ולדמן (talk) 12:28, 2 June 2024 (UTC)[reply]

I suppose that you mean to define

p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},

where the

a_{i}

are integers, and hope to derive a contradiction from the assumption

p(\theta )=0.

For that, doesnt'it suffice to show that the value of the integral is non-zero?

I'm afraid I'm not the right person to judge whether this approach offers a glimmer of hope. --Lambiam 15:40, 2 June 2024 (UTC)[reply]