CMC - Problem 6

Problem 6. (7 points) Find the smallest prime factor of $4294967297$ .

#NumberTheory #CMC #PointsExchange #MathProblem #Math

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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}`	$\boxed{123}$

Comments

This is pretty silly. This is the 5th fermat number, which is $2^{32}+1$ . It can be written as $2^{28}(5^4+2^4)-(5 \cdot 2^7)^4+1=2^{28} \cdot 641 -(640^4-1)$ , so it's clear that 641 divides it. For another prime that divides it, we must have $2^{32} \equiv -1 \pmod{p}$ , and therefore $2^{64} \equiv 1 \pmod{p}$ , so by Fermat's Little Theorem we must have that 64 divides p-1, or that p=64k+1. Now there are 9 possibilities, but some of them aren't prime. It's easy to check them.

Patrick Hompe - 7 years, 6 months ago

+1 for the first sentence.

Ahaan Rungta - 7 years, 6 months ago

6 points for Patrick Hompe! -7 points for calling it silly! >:O Did you know that in general, each prime factor of $F_{n\ge2}$ is in the form of $k\cdot2^{n+2}+1,\text{ }k\in\mathbb{N}$ ?

Cody Johnson - 7 years, 6 months ago

641

Ahaan Rungta - 7 years, 6 months ago

Python:

def primes(n):
    if n<=2:
        return []
    sieve=[True]*(n+1)
    for x in range(3,int(n**0.5)+1,2):
        for y in range(3,(n//x)+1,2):
            sieve[(x*y)]=False

    return [2]+[i for i in range(3,n,2) if sieve[i]]

Ahaan Rungta - 7 years, 6 months ago

Euler showed that this is the first composite Fermat number :D 641 divides it

Also, all the prime factors of $a^{2^n}+b^{2^n}$ are of the sort $1+k.2^{n+1}$

Bogdan Simeonov - 7 years ago

Why does this have the "points exchange" tag? Also, is this possible to do without W|A? If somebody finds it without W|A, it's pure luck. Also, fun fact: my FTC Robotics team ID number is 7297.

Ahaan Rungta - 7 years, 6 months ago

Yes, there's a brilliant solution to this. The answer is 641, so you get the 1 point.

Cody Johnson - 7 years, 6 months ago

the 1 point or 7 points?

Michael Tang - 7 years, 6 months ago

@Michael Tang – Look at the updated rules.

Cody Johnson - 7 years, 6 months ago

"Brilliant solution"

Lol pun

Daniel Wang - 7 years, 6 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}`	$\boxed{123}$