RESOLVED (FALSE): DOUBLING THE CUBE - 430 BC
In the spirit of the Parallels posture, the very ancient problem of Doubling the Cube - might at a stretch be called a conjecture - it goes back to the ancient Greeks, with numerous attempts to solve it.
I.e. many people at the time thought that it must be possible to construct a cube that is double the volume of an existing cube using ruler and compass methods - and to this day crank mathematicians continue to atempt it.
Doubling the cube
If Plutarch's account of the Delphi oracle and doubling the volume of the oracle is correct, then that dates it back to 430 BC. So a bit before Euclid at 300 BC approx.
Finally proved to be impossible in 1837
"Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas" - by Pierre Wantzel
RESOLVED (FALSE): PYTHAGORAS IDEA THAT EVERY MEASUREMENT IS RATIONAL 570 BC
In a similar vein there's the proof attributed to Pythagoras (born 570 BC) that root two is irrational - which caused major problems for their aim to reduce everything to pure numbers and ratios.
So that was a conjecture (that all measurements are rational) that got disproved fairly early on.
If you count that one, it goes back a bit further than the cube doubling.
(suggested by John Bailey in comment)
UNRESOLVED: GOLDBACH, 1742
If though you mean - oldest mathematical conjecture that is still a conjecture, neither proved nor disproved, well Goldbach's conjecture that every even number greater than 2 is a sum of 2 primes, it is pretty old, goes back to 1742
Goldbach's conjecture
It is still unsolved.
I don't know if it is the oldest ever.
Fun quote from Carl Sagan here
“Occasionally, I get a letter from someone who is in “contact” with extraterrestrials. I am invited to “ask them anything.” And so over the years I’ve prepared a little list of questions.
The extraterrestrials are very advanced, remember. So I ask things like, “Please provide a short proof of Fermat’s Last Theorem.” Or the Goldbach Conjecture. And then I have to explain what these are, because extraterrestrials will not call it Fermat’s Last Theorem. So I write out the simple equation with the exponents.
I never get an answer. On the other hand, if I ask something like “Should we be good?” I almost always get an answer.”
Quotes About Goldbach Conjecture
RESOLVED (TRUE) BUT ONLY RECENTLY, FERMAT 1637
Fermat's Last Theorem (1637) is older but of course
- It was stated as a theorem originally (though many mathematicians doubt that he solved it)
- It has now been proved.
UNRESOLVED: TWIN PRIMES - ENC. BRIT. SAYS 300 BC BUT NO CITATION - EARLIEST CITED REF, 1849
You might think the Twins Prime conjecture was a candidate - this is the conjecture that there are infinitely many primes of the form p, and p+2, only two apart.
On the face of it, it seems simpler, and Euclid proved that there are infinitely many primes.
Encyclopedia Britannica attributes this to Euclid, which would make it about 300 BC.
Greek mathematician Euclid (flourished c. 300 bce) gave the oldest known proof that there exist an infinite number of primes, and he conjectured that there are an infinite number of twin primes.Britannica doesn't include citations however to follow up to confirm this.
twin prime conjecture (number theory)
Here is a math overflow discussion, earliest mention that anyone turned up was a more general conjecture by Polignac in 1849, which implied the twin prime conjecture.
Twin Prime Conjecture Reference
Anyone know of any evidence that Euclid did make the twin prime conjecture - or know of any other unsolved conjectures from before 1742
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