I remember reading in Elon Musk: Tesla, SpaceX, and the Quest for a Fantastic Future something that Elon had been formally admitted to the PhD program at Stanford, but dropped out after three days. Vance contacted the Registrar's Office (or the graduate department itself?, I cannot remember) and received the confirmation that he was admitted.

He wanted to work on super capacitors in his PhD, but he realized that somebody would work on it anyways, and it's better for him to work on Paypal.

Watch the video of Sal Khan(fron Khan Academy) interviewing Elon musk.

Mostly right :). He actually thought it better to be involved in creating an internet company. This led to the founding of Zip2 (a kind of precursor to Yelp whose pivots ended up trying to help newspapers get online). The money from that led to the founding of X (think, transaction), which later took over and ultimately merged with its competitor, PayPal.

I second the recommendation of watching the interview. There is way too much misinformation surrounding Tesla and Elon Musk.

Yes, I recall him saying something similar. I was writing from my memory.

I couldn't find anything either.

My guess would be that you aren't technically in the PhD program until you've passed your qualifying exam. Until that point, you're a graduate student but not a PhD student. After you pass your preliminary exam, you're a PhD/Doctoral candidate.

Also, it's a moot point. There are no bragging rights to be found in simply being admitted to a PhD program. Plenty of very intelligent people fail to complete a PhD program (usually, IME, because of personality issues/conflict with the advisor (sadly extremely common), inflexibility of the advisor and/or department, or personal reasons (economic, social, health)).

(to be clear, Musk didn't brag that he was admitted to Stanford - he suggested that he felt dropping out was more useful than continuing. His fans often hold up being admitted to a PhD program as evidence of his engineering background)

In the USA, this is how it typically works. People who are enrolled in PhD programs are called "PhD students." Once they pass their advancement to candidacy exam, they are called "PhD Candidates." In both cases, they are enrolled in a PhD program. Typically people are accepted directly to PhD programs and they are not just admitted as generic graduate students.

The biography of Elon Musk by Ashlee Vance talks about this. Which I recommend everybody read. I don't remember the details but I think he was admitted but not enrolled. He ended up not going because he thought pursuing an internet company in Silicon Valley was more valuable than research work at the time.