In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object, such as a number, function, algorithm, or even proof.
Math History – Conjectures Proven Solvable Without Known Direct Proof
conjecturesmath-historyprovabilitysoft-question
Related Solutions
Weierstrass' famous function disproved a lot of results.
Dunham1 reports that:
The renowned Andre-Marie Ampere had presented a proof that continuous functions are differentiable in general, and calculus textbooks throughout the first half of the nineteenth century endorsed this position... [The Weierstrass function] not only refuted Ampere's "theorem" but drove the last nail into the coffin of geometric intuition as a trustworthy foundation for the calculus.
Dunham lists several other examples (Cauchy disproving Lagrange's formulation of calculus in terms of Taylor series, Dirichlet finding a function which can't be written as a Fourier series) but I'm not enough of a historian to say if these had a "large effect" on results.
Edit: you might find this post interesting. E.g. this excerpt taken from an article by Erica Klarreich July 20, 2009 on the Simons Foundation website.
In the 1970s and 1980s, mathematicians discovered that framed manifolds with Arf-Kervaire invariant equal to 1 - oddball manifolds not surgically related to a sphere - do in fact exist in the first five dimensions on the list: 2, 6, 14, 30 and 62. A clear pattern seemed to be established, and many mathematicians felt confident that this pattern would continue in higher dimensions…Researchers developed what Ravenel calls an entire cosmology of conjectures based on the assumption that manifolds with Arf-Kervaire invariant equal to 1 exist in all dimensions of the form $2^n−2$. Many called the notion that these manifolds might not exist the Doomsday Hypothesis, as it would wipe out a large body of research. Earlier this year, Victor Snaith of the University of Sheffield in England published a book about this research, warning in the preface, …this might turn out to be a book about things which do not exist. Just weeks after Snaith’s book appeared, Hopkins announced on April 21 that Snaith’s worst fears were justified: that Hopkins, Hill and Ravenel had proved that no manifolds of Arf-Kervaire invariant equal to 1 exist in dimensions 254 and higher. Dimension 126, the only one not covered by their analysis, remains a mystery. The new finding is convincing, even though it overturns many mathematicians’ expectations, Hovey said.
- Dunham, William. The calculus gallery: Masterpieces from Newton to Lebesgue. Princeton University Press, 2005.
Too long for a comment : I believe that its origins will be forever lost in the mist of time,
for the following very simple reason :
Euler was already aware in the eighteenth century that all primes except for $2$ and $3$ are of
the form $6n\pm1.~($At any rate, such a trivial statement is relatively easy to either discover
or understand, even by people with only the most basic mathematical knowledge$).~$
Then the next question which naturally arises is about the density of those “lucky” values of
n for which both neighbors of $6n$ are simultaneously prime. So basically all that's left to do,
after first dispensing with certain formalities pertaining to what is considered to be academi–
cally acceptable mathematical etiquette, such as actually proving that their number is indeed
infinite $($most likely by using some painfully obvious argument based, say, on reduction to
the absurd, and the like : as in the case of proving that there are an infinite number of primes,
for instance$),~$ would be getting down to the really hard part of actually quantifying their
frequency, and then venturing to offer a mathematical explanation for the experimentally
obtained results $\ldots$
Except that —oh, wait a second— remember that first “easy” half we were talking about just
earlier ? Well, as “luck” would have it, it turned out to be not so easy after all $\ldots$ So that's it,
in a nutshell.
Best Answer
It is known that there is an even integer $n\le246$ such that there are infinitely many primes $p$ such that the next prime is $p+n$, but there is no specific $n$ which has been proved to work (although everyone believes that every even $n\ge2$ actually works).