The Fibonacci numbers are defined as follows: $f_0=0$, $f_1=1$, and $f_n=f_{n−1}+f_{n−2}$ for $n≥2$. Prove each of the following three claims: (i) For each $n≥0$, $f_{3n}$ is even. (ii) For each $n≥0$, $f_{3n+1}$ is odd. (iii) For each $n≥0$, $f_{3n+2}$ is odd. Solution (i) Base case: $n=0$, $f_{3(0)}=f_0=0$ (it is even) (ii) Base case: $n=0$, $f_{3(0)+1}=f_1=1$ […]

I’m looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously): $ \sigma_0(n,m) = \sum_{d|n,d<m} 1 $. Or more generally: $ \sigma_x(n,m) = \sum_{d|n,d<m} d^x $. Or any important property of either (like […]

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 […]

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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