×

# Prime Sequence

Consider the sequence $$a_1 = 101, a_2 = 10101, a_3 = 1010101$$ and so on. Prove that $$a_k$$ is composite iff $$k\geq 2$$.

Note by D K
1 year, 8 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Aha I see it. Any even one a sub 2k can be written in the form (a sub k)^2-100^k which neatly factors to (a sub k+10^k)(a sub k-10^k) for all evens. Sorry for my latex. And the odd case is trivial.

- 1 year, 8 months ago

I have another unanswered discussion question. Can you take a look at it?

- 1 year, 8 months ago

Nice observation. Kudos.

- 1 year, 8 months ago

That's interesting. I can show that if $$k$$ is composite, then $$a_k$$ is composite and has a factor of 101...101.

- 1 year, 8 months ago

you mean if k is odd ? then ak will have 101 as a factor. but I also need proof for when k is even.

- 1 year, 8 months ago