The story is told in some detail in Sylvia Nasar's "A Beautiful Mind", a biography of John Nash, who was a fellow student of Milnor at one point. In that version, Milnor knew that Borsuk's conjecture was an open problem; he wrote up his apparent answer not believing it to be correct, and asked Fox to look it over since he (Milnor) hadn't been able to find the error himself. Fox told him to write up the result for publication; the final result was generalized considerably over the original version. It's pretty likely that Nasar interviewed Milnor (because of his biographical connection with Nash) while writing the book, so her version is probably as good as you'll find.
The "came to class late and thought it was a homework problem" story is about George Dantzig and is easy to find on the internet (e.g. Wikipedia or Snopes). It was about some problem in statistics. I think there may have actually been two open problems involved.
Sometimes the Dantzig story gets told with SIX open problems. That might be confabulated with Grothendieck's PhD thesis. Dieudonné and Schwartz had written a paper on functional analysis ending with six apparently difficult open problems. They turned these over to Grothendieck saying something like "see if you can make some progress on any of these, and that can be your thesis." Within a few months Grothendieck developed the theory of nuclear spaces, that turned all six problems into trivial calculations, and basically killed off the research direction originally proposed (by completely solving it). That story is from Allyn Jackson's biographical profile of Grothendieck in Notices of the AMS, I think.
The modern reference work on the subject seems to be [1], but it spends only a page and a half on the subject of primes in multivariate quadratic polynomials (pp. 396-397). More than half this space is devoted to Iwaniec's 1974 result. The balance mentions Sarnak's application to the Problem of Apollonius and a result of "J. Cho and H. Kim" on counting primes in $\mathbb{Q}[\sqrt{-2}].$ So nothing there.
Pleasants [2] shows that, subject to a Davenport-Lewis [3] condition on the $h^*$ (a complexity measure on the cubic form part), multivariate cubic polynomials have the expected number of primes. Unfortunately this condition requires (as a necessary but insufficient condition) that there be at least 8 variables. Further, it double-counts repeated primes.
Goldoni [4] recently wrote a thesis on this general topic. His new results (Chapter 5) on the $h$ and $h^*$ invariants make it easier to use the results of Pleasants but do not extend them to cubic polynomials with fewer than 8 variables.
Of course I would be remiss in failing to mention the groundbreaking work of Heath-Brown [5], building on Friedlander & Iwaniec [6]. These results will no doubt clear the way for broader research, but so far have not been generalized.
So in short it appears that:
- Nothing further is known about primes represented by quadratic polynomials.
- Apart from $x^3+2y^3$, almost nothing is known about which primes are represented by cubic polynomials, though some results are known for how often such polynomials take on prime values provided $h^*$ and hence the number of variables is large enough.
On the historical side, of course Fermat is responsible for the proof of the case $x^2+y^2$. I have references that say that Weber [7] and Schering [8] handled the case of (primitive) binary quadratic forms with nonsquare discriminants, but I haven't read the papers. Motohashi [9] proved that there are $\gg n/\log^2 n$ primes of the form $x^2+y^2+1$ up to $n$, apparently (?) the first such result with a constant term. He conjectured that the true number was
$$\frac{n}{(\log n)^{3/2}}\cdot\frac32\prod_{p\equiv3(4)}\left(1-\frac{1}{p^2}\right)^{-1/2}\left(1-\frac{1}{p(p-1)}\right)$$
but as far as I know the constant still has not been proved even for this special form.
Edit: Apparently Bredihin [10] proved the infinitude of primes of the form $x^2+y^2+1$ some years before Motohashi. He only gave a slight upper-bound on their density, though: $O(n/(\log n)^{1.042}).$ (Motohashi improved the exponent to 1.5 in a later paper.)
[1] Friedlander, J. and Iwaniec, H. (2010). Opera de Cribro. AMS.
[2] Pleasants, P. (1966). The representation of primes by cubic polynomials, Acta Arithmetica 12, pp. 23-44.
[3] Davenport, H. and Lewis, D. J. (1964). "Non-homogeneous cubic equations". Journal of the London Mathematical Society 39, pp. 657-671.
[4] Goldoni, L. (2010). Prime Numbers and Polynomials. Doctoral thesis, Università degli Studi di Trento.
[5] Heath-Brown, D. R. (2001). Primes represented by $x^3 + 2y^3$. Acta Mathematica 186, pp. 1-84; Wayback Machine.
[6] Friedlander, J. and Iwaniec, H. (1997). Using a parity-sensitive sieve to count prime values of a polynomial. Proceedings of the National Academy of Sciences 94, pp. 1054-1058.
[7] Weber, H. (1882). "Beweis des Satzes, dass, usw". Mathematische Annalen 20, pp. 301-329.
[8] Schering, E. (1909). "Beweis des Dirichletschen Satzes". Gesammelte mathematische Werke, Bd. 2, pp. 357-365.
[9] Motohashi, Y. (1969). On the distribution of prime numbers which are of the form $x^2+y^2+1$. Acta Arithmetica 16, pp. 351-364.
[10] Bredihin, B. M. (1963). Binary additive problems of indeterminate type II. Analogue of the problem of Hardy and Littlewood (Russian). Izv. Akad. Nauk. SSSR. Ser. Mat. 27, pp. 577-612.
Best Answer
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/\mathbb{Q}$, both the size of the torsion subgroup of $E(\mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: Conjecture There are only finitely many $E/\mathbb{Q}$ for which the rank of $E(\mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)