Factoring a what?

This is a question from the International Mathematics Contest 2013, I've been trying to solve it for a long time, But I couldn't find a legitimate solution. Could anyone give a mathematical solution to it? Algebraic, Geometric (if something of the kind exists), Thanks! =D

Give a factor of the expression: $2^{42} + 2^{21} + 1$ .

#NumberTheory

Easy Math Editor

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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`	$\sum_{i=1}^3$
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`\boxed{123}`	$\boxed{123}$

Comments

Seems easy for a IMC Question, unless I messed up.

$2^{42} + 2^{21} + 1 = \frac{2^{63} - 1}{2^{21} - 1}$

So we need to find a prime for which $2^{63} \equiv 1 \pmod{p}$ but $2^{21} \not \equiv 1 \pmod{p}$ .

Since $9|63$ and $9\not | 21$ , if we find a prime such that $2^9 \equiv 1 \pmod{p}$ but $2^3 \not \equiv 1 \pmod{p}$ , we are done.

Now $2^9 - 1 = 511 = 7*73$ . So $2^9 \equiv 1 \pmod{7,73}$ . But, we have $2^3 \equiv 1 \pmod{7}$ and $2^3 \not \equiv 1 \pmod{73}$ . So 7 is ruled out. So the prime which we require is $73$ .

We can verify this since $2^{42} + 2^{21} + 1 = 2^{36 + 6} + 2^{18+3} + 1 \equiv 2^6 + 2^3 + 1 = 64 + 8 + 1 = 73 \equiv 0 \pmod {73}$

Siddhartha Srivastava - 5 years, 4 months ago

Thank you so much for the response! =D But could I ask one more thing? I can see you used Fermat's little Theorem there, but I don't understand how you factored the first equation. I'd really appreciate it if you could explain that part too. Thaaaanks =DD

EDIT: Or wait, is it Fermat's little theorem??? XDD

Giwon Kim - 5 years, 4 months ago

$2^{42} + 2^{21} + 1 = \frac{2^{63} - 1}{2^{21} - 1}$

This equation?

It's just some simple algebra. $x^2 + x + 1 = \dfrac{(x^2 + x + 1)(x-1)}{(x-1)} = \dfrac{x^3 - 1}{x-1}$ , where $x = 2^{21}$

Siddhartha Srivastava - 5 years, 4 months ago

@Siddhartha Srivastava – Ohhhh I see, XD The idea is quite simple, but I've never encountered it before. That really helped a lot, XDD Thanks!

Giwon Kim - 5 years, 3 months ago

Math	Appears 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}`	$\boxed{123}$