What is a First Word™?

Question

_{This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.}

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If a word conforms to a special rule, I call it a First Word™.
Use the examples below to find the rule.

$$ % \def\Pad{\P{0.0}} \def\Title{\textbf{ First }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Words}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ } \smash{\lower{29px}\bbox[lightblue]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ first}&\text{ third}\\ \hline \text{ saturn}&\text{jupiter}\\ \hline \text{ dog}&\text{turtle}\\ \hline \text{ put}&\text{out}\\ \hline \text{ washington}&\text{paris}\\ \hline \text{ red}&\text{black}\\ \hline \text{ thought}&\text{ though}\\ \hline \text{ cork}&\text{ cook}\\ \hline \text{ conundrum}&\text{ puzzle}\\ \hline \text{ astonishment}&\text{ surprise}\\ \hline \text{ bar}&\text{foo}\\ \hline \text{ invalid}&\text{valid}\\ \hline \text{ auto}&\text{mobile}\\ \hline \text{ plot}&\text{agitates}\\ \hline \hline \end{array}$$

CSV version:

First Words™,   Not First Words™

first,          third
saturn,         jupiter
dog,            turtle
put,            out
washington,     paris
red,            black
thought,        though
cork,           cook
conundrum,      puzzle
astonishment,   surprise
bar,            foo
invalid,        valid
auto,           mobile
plot,           agitates

What is the special rule these words conform to?

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Hint:

Trivially, pizza, apple, macaroni, burrito, tofu are not First Words. However, ale is one. Why is that?

Answer 1

Update: Got it at last:

first = prime

Write 1 for consonants and 0 for vowels, read as binary number and check whether the result is prime.