FINAL EDIT : Prove that if $p^z|n^2-1$ $$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn’t hold for any chosen values of $p,x,n$ and $z$. Here $p>3$ is an odd prime , $x=2y+z, \ \{\{x,y,z\}>0\} \in \mathbb{Z}$ . There $n$ is an even number. If the above statement is prove it will lead to a contradiction$^*$ $^*$: to understand the contradiction […]

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn’t find anything. Thanks!

Some books say even numbers start from two but if you consider the number line concept, I think zero should be even because it is in between -1 and +1 (i.e in between 2 odd numbers). What is the real answer?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.

Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value? ($p\lt n$, $p$ and $n$ are positive integers)

Plato puts the following words in Socrates’ mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called odd, which is not the same as three? […]

Lotus leaves are arranged around a circle. A Frog starts jumping from one leaf in the manner described below. In the first jump it skips one leaf,next jump it skips two,three the next jump and so on. If the frog can reach all the leaves, show that number of leaves cannot be odd.

This question already has an answer here: Proving that an integer is even if and only if it is not odd 2 answers

Is the concept of even/odd numbers is applicable to decimal numbers? For e.g. – 4.222 is a even number?

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no other piece in the game. Background: The Tetris playing field has width $10$. Rotation is allowed, so there are […]

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