Show using induction that for every $n\in \mathbb{N}$ the number $P_n=2^{2^{n+1}}+2^{2^n}+1$ is divisible by $21$.

Question

So the first step is to check for $n=1$ $$P_1=2^{2^2}+2^2+1=16+5=21,$$ which is divisible by $21$, so the statement holds for $n=1$. Let for some $n=k\ge1,P_k=2^{2^{k+1}}+2^{2^k}+1$ is divisible by $21$. Now we should try to prove that the statement also holds for $n=k+1$, or show that $P_{k+1}=2^{2^{k+2}}+2^{2^{k+1}}+1$ is divisible by $21$. We can write $P_{k+1}$ as $$P_{k+1}=2^{2\cdot2^{k+1}}+2^{2\cdot2^k}+1$$ How do we use the induction hypothesis?

One way would be to use induction instead on what $2^{2^k}$ is modulo $3$ and modulo $7$. — John Omielan, Feb 01 '22 at 21:13
@JohnOmielan, I haven't studied the concept of modulo, and don't think that I am allowed to use it. — kormoran, Feb 01 '22 at 21:15
The modulo, as I stated in my comment, just means the remainder after dividing. — John Omielan, Feb 01 '22 at 21:18
FYI, I couldn't find asking to prove your expression is divisible by $3$ or $21$, but $5$ questions asking to prove it's divisible by $7$ are Prove by induction that $1+2^{2^n}+2^{2^{n+1}}$ is divisible by $7$, Proof for $\forall n\in \mathbb{N}: 7\mid(1 + 2^{2^n} + 2^{2^{n+1}})$, Divisibility by 7 Proof by Induction, $4^{2^n}+2^{2^n}+1$ is Divisible by $7$ and Prove that $7$ divides $1 + 2^{(2^n)} + 2^{(2^{n+1})}$ by induction. — John Omielan, Feb 01 '22 at 21:34

score 3 · Accepted Answer · answered Feb 01 '22 at 21:11

3

Let $A_n=2^{2^n}$. Then $P_n=2^{2^{n+1}}+2^{2^n}+1=A_n^2+A_n+1$ and...

$$P_{n+1}=2^{2^{n+2}}+2^{2^{n+1}}+1=2^{4\cdot2^n}+2^{2\cdot2^n}+1=A_n^4+A_n^2+1\\=(A_n^2+A_n+1)(A_n^2-A_n+1)=P_n(A_n^2-A_n+1)$$

And $P_1=21$. So... the answer is clear! If $21\mid P_n$, then automatically $21\mid P_{n+1}$ holds!

answered Feb 01 '22 at 21:11

MH.Lee

5,568

Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Apr 03 '23 at 00:21

Show using induction that for every $n\in \mathbb{N}$ the number $P_n=2^{2^{n+1}}+2^{2^n}+1$ is divisible by $21$.

1 Answers1