A theorem for fun

[Abdul Muntakim Rafi]

I have recently found this theorem while solving a problem... I was trying to solve the problem... I was thinking about different cases... And I came up with the theorem... Can you prove it?


\[(2a+1)^{2b+1}\equiv 2a+1 (mod 2c)\]

and \[(2a+1)^{2b}\equiv 1 (mod 2c)\]
which is the same thing as the above one...

Here $a,b,c$ are non negative integers... The theorem is true for some non negative values of
$a,b,c$ ...
I think if we find a way to determine what the values of $a,b,c$ are this might be considered as a theorem...

[*Mahi*]

\[(2a+1)^{2b+1}\equiv 2a+1 (mod 2c)\]

and \[(2a+1)^{2b}\equiv 1 (mod 2c)\]

You say something like this is a theorem? Like solving some congruence like $x^y \equiv 1 (mod z)$ where $x$ is odd and $y,z$ are even? It might be a problem, but a theorem consists of much more things. And there is much in fact, to solve conguences like $x^y \equiv 1 (mod z)$.

[afif mansib ch]

am i getting the thing wrong or it is easy?i think i'm wrong with something

we know for any constant a and c
\[(2a)\equiv0(mod2c)\]
and (2a+1)^(2b)is congruent to 1(mod 2c)
now multiplying both sides by (2a+1) we get
(2a+1)^(2b+1)is congruent to 2a+1(mod 2c)
is it all?
[i am really sorry to write down like this.i am not being able to learn using latex properly.so i wrote like this ]

[Abdul Muntakim Rafi]

Well that's why I called it a theorem for fun... You can see it as a problem...

And Afif, first you have to prove that
(2a+1)^(2b) is congruent to 1(mod 2c)
then we are done...
By the way,I used this theorem(!) while solving your congruence problem...

[*Mahi*]

we know for any constant a and c
\[(2a)\equiv0(mod2c)\]

Is it? What about $a=3,c=4$ and infinite more $a,c$

[afif mansib ch]

sorry mahi via i didn't notice it. watching 2 in both sides i wrote that. then what should i write?

[afif mansib ch]

i wanted to say if one of the statement is true by multiplying or dividing it by (2a+1)we can get the other one.so what else is needed to be proved?

[*Mahi*]

Okay you might need to know some of the basics about congruence, so please see this post, and try some textbooks.

[afif mansib ch]

i am act reading the book ELEMENTARY NUMBER THEORY.thanks for the suggestion.have i got the ques wrong?i think he meant to proof if one of the statement is true then the other one is also.so do i need to proof anything else?

[*Mahi*]

I think you do have the question wrong...it's about just some cases of when those two things would happen.
BTW you may also try Number Theory by S.G.Telang as a text, it is also good for beginning problem solvers like you.