I will try to prove that such an odd prime p doesn't exist

$2^p -1$ is divisible by 3 if and only if the sum of its digits is divisible by 3

now

$2^p mod 3 = 2^( [p mod 2] ) mod 3 $ (2 is the totient of 3)

and this equals 1 only if $p mod 2 = 0 $

therefore, an odd prime p doesn't exist

Thanks for your answer

I proved without totient that 2^p -1 is not divisible by 3 if p is an odd number so I asked my question above

Peter Breuer Please stop arguing with me. You defend the totient Function like if you are Euler himself. I even changed the question of this discussion. Actually there is a sequence related to the Collatz conjecture which can help to define a set of numbers which has the cardinal of the set of even numbers. How do you see the usefulness of this sequence?

Peter Breuer You say in your profile that you are a senior science Officer. Does it mean that I am the science criminal who proved to the world that your country is lying to the world about the illuminati monk who you call Einstein ?

Peter Breuer What's your purpose ? you are following like if you are my wife. Go find some love in another place and not in my discussion.

Peter Breuer I am a better researcher and scientist than you. At least I don't lie to myself and to the world that there are spacetime curvatures around us. You are a delusional lover of the lies of the illuminati who refuses the pure truth that our spacetime is an obvious Euclidean + time dimension.

Peter Breuer You don't publish since 2016 whereas I don't stop working. I present to you my new article that proves that Theta-3 Functions give only irrational outputs when the inputs are rationals. This is a continuation to my works about Theta Functions by using a therorem of Dirichlet. I wait for your collaboration. Here is the link:

https://www.researchgate.net/publication/377658268_The_proof_that_the_rational_numbers_as_input_values_in_Theta-3_Functions_give_only_irrational_numbers_as_output_in_these_functions

Peter Breuer Theta Functions are transcendental functions however their output values are not all transcendental. Hence it is not proved that Theta Functions give only irrational outputs. If Theta-3 Functions give only irrational outputs then this will contradict their continuity. Who should revise his lessons my boy?

Peter Breuer The wikipedia article says only that the Theta Function output is transcendental for the lemniscate values and not algebraic values. Furthermore, Theta Functions are not Rogers-Ramanujan continued fraction. What's up with you Peter ?

Peter Breuer You can't express all rationals as Lemniscatic values. Do it if you can and share the results with us. Furthermore, you will find attached an image from Wikipedia expressing Rogers-Ramanujan continued fraction in terms of Theta Functions.

Peter Breuer we agree about the Wikipedia content when it says that the lemniscatic values of theta-3 functions are also transcendental. It is clear there that lemniscatic values are the outputs of the Theta-3 functions and not the lemnastic values of input of Theta-3 functions.

A lemniscatic value is transcendental because it is any value that can be written in terms of the Lemniscate Constant which is transcendental. Try to be at least open-minded my boy because before you started discussing all the audience agreed that Theta-3 Functions can't be irrational thanks to its proven continuity. At least ask your scientific friends about it, what's wrong with you?

Here is the link attached:

https://en.wikipedia.org/wiki/Lemniscate_constant

Peter Breuer You are mistaken. There is only one constant called "The Lemniscate Constant" (not Lemniscatic constant) and this constant acts in algebra just like Pi acts in geometry. Both Pi and "The Lemniscate Constant" are transcendental.

Now, a "Lemniscatic value" is any value expressed in an algebraic manner by using "The Lemniscate Constant".

And thus, the meaning of the sentence in Wikipedia is about some possible "Lemniscatic values" as outputs because they are possible as outputs of Theta Functions since these functions can be linked in an algebraic manner to Rogers-Ramanujan continued Fraction.

Can you go and ask a respected mathematics professor who you trust about this issue instead of arguing here like a Cowboy???

Peter Breuer Yes I mean that "Lemniscatic value" is rational function of roots and powers of any "Lemniscate Constant" that you choose. I am telling you again that the Wikipedia content says only that a "Lemniscatic value" can be an output of Theta Functions. Remember that thanks to the continuity of Theta Functions which is a well establihed fact, we can never accept that Theta Functions give only Transcendental or irrational output values.

Peter Breuer A Lemniscatic value is transcendental and thus irrational. However, not all output values of Theta Functions are Lemniscatic values. Hence, not all outputs of Theta Functions are irrational.

Peter Breuer I could read the document. The issue concerns only rational numbers. I proved that for q is a rational number with 0

Peter Breuer As a definition: The algebraic closure of the rational numbers, denoted as Q-bar, is the set of all algebraic numbers, which are the roots of non-constant polynomials with coefficients in Q.

Peter Breuer As I said, the real numbers which are transcendental are not included in the algebraic closure of the rational numbers denoted Q-bar.

Peter Breuer We were discussing the issue of rational numbers. You said that the outputs of Theta-3 Functions are always irrational whatever is the input and that's false because the transcendental numbers as inputs in Theta-3 Functions may give rational outputs. This is a result of the continuity of Theta-3 Functions.

Peter Breuer if you only meant that then why were we arguing about Lemniscatic values ???

Nour-Eddine Rahmani I published a new short mathematical letter about the Collatz conjecture which is related to my previous article in this topic. This short article gives the only possible five expressions of natural numbers which can respect the Collatz conjecture. Can you now find the counterexample by using the results or prove that the Collatz conjecture is true ?

Here is the link:

https://www.researchgate.net/publication/377746599_The_only_five_expressions_of_numbers_which_respect_the_Collatz_conjecture