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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.

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 $k\cdot2^{n+2}+1,\text{ }k\in\mathbb{N}$?

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]]

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.

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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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## Comments

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TopNewestThis 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.

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+1 for the first sentence.

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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 $k\cdot2^{n+2}+1,\text{ }k\in\mathbb{N}$?

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641

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Python:

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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 $1+k.2^{n+1}$

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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.

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

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the 1 point or 7 points?

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"Brilliant solution"

Lol pun

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