In the first Harry Potter book, it is said that Flamel and Dumbledore invented the philosopher's stone together. It's also heavily implied that this stone is the only reason why Flamel lived so long. This means that Dumbledore is… let's say less than 200 years younger than Flamel, to be safe.
But since Flamel's age is given and is somewhere in the 600s, that would mean that Dumbledore is more than 400 years old. But after the stone was destroyed, he did not die from that and it's also not said that he had a stash of the elixir.
In book 5 there's even a short appearance of someone who was an examinant while Dumbledore was still in school, meaning she would have to be older than him. I would be very surprised if she also got the elixir of life.
The same with Dumbledore's brother: Did he get the elixir? Or their parents and he's multiple centuries younger?
Or are there other ways that wizards become many centuries old? In that case I would wonder why an elixir of life was even necessary.

The only explanations I can think of are these:

  • Dumbledore wasn't actually involved in creating the stone and book 1 lied.
  • Flamel isn't actually that old and book 1 lied (and maybe others as well).
  • Rowling is very bad at Math, much more than usual.
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    Where does it say that Dumbledore was involved in the development of the Stone or the Elixir of Life? – Valorum Aug 1 '19 at 19:37
  • @Valorum On the back of his chocolate frog card. – Fabian Röling Aug 1 '19 at 19:51
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    @FabianRöling - "Albus Dumbledore, currently Headmaster of Hogwarts. Considered by many the greatest wizard of modern times, Professor Dumbledore is particularly famous for his defeat of the dark wizard Grindelwald in 1945, for the discovery of the twelve uses of dragon’s blood and his work on alchemy with his partner, Nicolas Flamel. Professor Dumbledore enjoys chamber music and tenpin bowling." - No mention of him working on the Stone, just collaborating with Flamel on alchemy – Valorum Aug 1 '19 at 19:52
  • I just looked it up again in the German book that I read, in case there was a mistranslation. And yes, it is actually translated in an unexpected way, but not like that. It says "… und auf seinem Werk über Alchemie", where "Werk" means "a work", like a book or similar. But it does in fact not mention the philosopher's stone at all. BUT in the chapter "Nicolas Flamel" (I assume it's the same in English), they read in a book something like "alchemy is the science of creating the philosopher's stone" (I don't know the exact wording in English), which is ridiculous, but would imply that Dumbledore – Fabian Röling Aug 1 '19 at 20:04
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    Alchemy isn't simply the process of creating the Philosopher's Stone (a mythical substance that supposedly confers eternal life and allows the transmutation of lead into gold) but more general sciencey things. – Valorum Aug 1 '19 at 20:06

Dumbledore was not involved at all in the creation of the Philosopher's Stone, and to my knowledge it is never implied that he was. He certainly was a good friend of Nicolas Flamel and worked with him on alchemical projects, but only centuries after Flamel created the Stone. Dumbledore was born in 1881, whereas Flamel created the Stone some time in the 14th century.

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