The US Open champion climbed to the summit of the ATP rankings in late February of this year and had a fortnight there before surrendering the top rung to Djokovic, the long-time incumbent.
Djokovic remains there and has spent a record 371 weeks at No.1 across his career, while Roger Federer had 310 weeks on top and Rafael Nadal 209.
Their dominance has been at the extreme end of the sporting spectrum, and Medvedev will probably never get close to matching any of their totals, but there is a strong chance he will jump back to No.1 sooner rather than later.
Medvedev revealed on Sunday (AEST) he had searched on the Internet to learn which players had short runs at No.1 comparable to his own, such is his fascination.
He is closely tracking Djokovic in the current world rankings, and should he reach the French Open final next weekend he will overtake the 35-year-old Serbian – even if Djokovic successfully defends his Roland Garros title.
Then the decision by the ATP to strip Wimbledon of ranking points means Djokovic will lose 2,000 points after that tournament, having been champion last year, while Medvedev has only a handful to hand back. He is banned from Wimbledon, as all Russians are, and is unhappy to be missing out, but he stands to benefit on the rankings list.
"I for sure want to be there more than two weeks. I want to try to do it," Medvedev said. "We don't know how, this Wimbledon thing, but I want to try to make the best results possible."
Medvedev recalled being told by Tennis Channel how he could return to No.1 in Paris, saying that was "great to know".
"That's great motivation," Medvedev said. "It's not something that pressures me, because I'm really happy I managed to do it.
"I remember at Indian Wells I lost, I didn't like my match against Gael [Monfils], and that's when I knew I was going to lose the number one spot.
"I was like, well, just two weeks. I went on the internet and looked just like this on Google, who were the shortest number one players in the world, and the first I saw was Pat Rafter being there for one week, and he's an absolute legend, and Carlos Moya was somewhere there. Two weeks.
"If somebody would ask me, how long were Rafter and Moya number one, I'd say, I don't know, six months, one year."
After a 6-2 6-4 6-2 win over Djokovic's compatriot Miomir Kecmanovic, which set up a Round of 16 clash with Maric Cilic, the 26-year-old Medvedev spoke of his pride at having briefly lorded it over his tour rivals.
"It's something nobody can take away from me," he said. "Not that many players could take it while they play tennis. It's a great motivation to try to come back there."