Fermat 'Prime' Mistake

Fermat Prime $latex F5= 2^{2^{5}}+1 $ composite ?

Proof: 1937 the 'Calculator boy' Zerah Colburn observed:

$latex 641 = 2^4 + 5^4 \mbox {..[1]} $
$latex 641 = 5.2^7 +1 \mbox {..[2]}$
$latex \mbox {[1:]}\: 2^4 + 5^4 = 0 \: mod \:641 $
Divide $latex 5^4 $ both sides
$latex \frac {2^4}{5^4} + 1 = 0 \:mod \:641$
$latex \frac {2^4}{5^4}= -1 \:mod \:641\mbox {..[a]} $

$latex \mbox {[2:]}\: 5.2^7 +1=0 \: mod \:641 $
$latex 2^7 = -1/5 \: mod \:641 $
x2 both sides
$latex 2^8 = -2/5 \: mod \:641 $
Raise power 4 both sides
$latex (2^8)^4 = (-2/5)^4 \: mod \:641 $
$latex 2^{32} = +2^4 / 5^4 \: mod \:641 $
$latex \mbox {[a:]}\: \frac {2^4}{5^4}= -1 \: mod \:641 $
$latex 2^{32} = -1 \: mod \:641 $
$latex 2^{32} +1= 0 \: mod \:641 $
$latex F5= 2^{2^{5}}+1 \mbox \: {divisible \:by \:641} $
=> $latex \mbox {F5 not prime!} $

Note:
$latex F1= 2^{2^{1}}+1 = 5 (prime)$
$latex F2= 2^{2^{2}}+1 = 17 (prime)$
$latex F3= 2^{2^{3}}+1 = 257 (prime)$
$latex F4= 2^{2^{4}}+1 = 65,537 (prime)$

$latex F5= 2^{2^{5}}+1 = 4,294,967,297 = 641 \mbox{x} 6,700,417 (nonprime)$

Euler (in 1732) had proved mentally F5 was not a prime.