Mike Pence has file to run for POTUS. Donald Trump did so late last year. Presumably they will be on the same R ballot in at least some states.
Has it ever been the case that a former US President and his Vice-president have faced one another in the Primaries?
Prior to 1800 the VP was whoever was the runner-up in the presidential election, and so this question is effectively inapplicable to the pre-1800 era. Which wasn’t all that many electoral cycles.
Since in the modern era there is seldom a real primary involving a sitting president running for immediate re-election to a second consecutive term (e.g. Biden in 2024, Trump in 2020), the likely way for this to occur is when a one-term president runs for president again after having been defeated in his first attempt at a second consecutive term.
If Trump pulls off this feat in 2024 he will be only the second such US president after Grover Cleveland, who’s famous for little else than his own two-nonconsecutive-term presidency. And Cleveland’s first term VP (Thomas Hendricks) did not run for the Democratic nomination for president in Cleveland’s unsuccessful bid for reelection in 1888, nor his successful bid 4 years later in 1892.
In the era before FDR and the two-term limit there may well have been cases where the sitting president and one of his prior VP(s) faced off in the primaries for a failed bid at a third term. But I doubt it, and I’m too lazy to investigate 40-some election campaigns times 2+ parties…
So IMO Trump would be the first.
His death in 1885 was a real roadblock there.
A candy bar that is still made today was named after his daughter.
Well, in 1844, both Van Buren and Richard Johnson (the former VP) sought the Democratic nomination. But the nomination was by convention, not primary. (James Polk won the nomination, and the election).
Sure it is… wink
The real question here is, how many people are going to register as Republicans, just so they can vote for Pence, as a “Fuck you!” to Trump?
Because losing to Pence would absolutely drive Trump mad.
That would drive anyone mad.
You lost to The Blandest Man in America?
Suddenly, I’m tempted.
In the states with registration by party and party-registrant-only primaries, there are a lot of reasons to register tactically when one party or the other has an interesting primary your vote can influence and one doesn’t.
In 2024 it’s the Ds turn to have the boring anointment primary and the R’s turn to have the knock-down drag-out primary.
Then again, in the era of modern statistics and propaganda I would be very reluctant to register as an R just to vote for not-Trump in my state’s primary because if very many D’s did that, the news would spread “Large numbers of historical D voters are abandoning their screwed-up anti-American party for the wonderful patriotic Rs”
Which “momentum” will have a lot of influence with the low info casual voters that both parties really need to attract and motivate to carry the day.
IOW, don’t hand the enemy a convenient narrative to beat you with. They can invent enough on their own without you giving their lies a central core of truth.
How long have primaries been around?
I believe they started in the early 1900s. Strictly speaking, of course, the nominee is still selected by convention, but now most states having a binding primary election.
But if we’re looking for an example of a primary election ballot with both a former president and his former VP on the ballot, it’s going to probably have to be post-1900.
John Nance Garner is the main example. He was Franklin Roosevelt’s Vice President. There was a question whether Roosevelt would run for a third term in 1940. When Roosevelt wouldn’t declare his intentions, Garner declared he was running. Roosevelt eventually decided to run and won the nomination (and subsequent election). He replaced Garner as his running mate with Henry Wallace.
That famous candy bar, the candy bar called Baby Ruth, was named after Grover’s daughter, Ruth.
And now you know…the rest of the story.
Not quite Prez v Veep, but in 1912 there was President Taft, former Secretary of War to Teddy Roosevelt, being challenged by Teddy in the Bull Moose party.
Also not quite Prez vs. Veep, but Henry Wallace -FDR’s third-term VP, ran against Harry Truman, FDR’s fourth-term VP, in 1948.
A lasting memorial, as Ruth Cleveland had died 17 years before the candy bar was introduced. 
That’s the company’s story… but it came out while Babe Ruth was extremely popular, two decades after “Baby” Ruth had died. Hence the wink.
To continue the side-track: the story about the Baby Ruth candy bar being named after an actual “baby”, and not a person named Babe Ruth, was subject to lawsuits. At the time, no one seriously believed the story that the candy bar was not named after the baseball star; baby Ruth kind of just showed up as an excuse to use that name. In the end, Babe Ruth lost the right to use his own name on a candy bar, but here we are today.
And both Baby Ruth and Babe Ruth are still dead. Along with General Franco. I’m not sure what it all means. 
It means none of those people are going to be running for Pres in the Republican primaries. Just about everyone else will be.