Years ago, I read a story about a mathematician who found a numerical counterexample to some conjecture long believed to be true. He gave a talk during which he didn't utter a single word but simply wrote out the arithmetic computation on the board and, as I recall the narrative, he "sat down to thunderous applause".
Does anyone happen to know the story I'm talking about?
You are most likely referring to the 1903 presentation by American mathematician Frank Cole. The original false conjecture was that the 67-th Mersenne number $M_{67}:=2^{67}-1$ is prime, and it goes back to the preface to Mersenne's own Cogitata Physica-Mathematica (1644). However, Cole was already confirming rather than disproving, that $M_{67}$ is composite was shown indirectly by Lucas in 1876, using what is now known as the Lucas primality test. The dramatic account of Cole's presentation is due to Eric Temple Bell, who attended it. But although Bell's account is often repeated, and embellished, there are reasons to doubt its accuracy, see below.
Bell writes in his 1951 book Mathematics; Queen and Servant of Sciences:
At the October, 1903, meeting in New York of the American Mathematical Society, Cole had a paper on the program with the modest title On the factorization of large numbers. When the chairman called on him for his paper, Cole—who was always a man of few words—walked to the board and, saying nothing, proceeded to chalk up the arithmetic for raising $2$ to the sixty-seventh power. Then he carefully subtracted $1$. Without a word, he moved over to a clear space on the board and multiplied out, by longhand, $193,707,721\times 761,838,257,287.$ The two calculations agreed. … For the first and only time in record, an audience of the American Mathematical Society vigorously applauded the author of a paper delivered before it. Cole took his seat without having uttered a word. Nobody asked him a question.
Again according to Bell, when he wrote to Cole in 1911 to ask how long it had taken him to crack $M_{67}:=2^{67}-1$, Cole reportedly answered: "three years of Sundays." However, according to Corry's article:
Historians of mathematics tend to distrust the historical reliability of most of Bell’s accounts, and in this case there are good reasons to stick to this habitude. For one thing, the Bulletin of the American Mathematical Society records the talks presented at its meeting of December 31, 1903, in New York, including Cole’s, precisely with the name mentioned by Bell. The text is much more elaborate than simply two arithmetic operations whose results are equated, and it contains some interesting ideas about the importance of the result and about how Cole went about finding the factors involved in his calculation (more on this below). One may certainly agree that Cole deserved the standing ovation, and indeed the ovation may have actually taken place. None of this, however, is mentioned in the Bulletin. As for the amount of time spent on the calculation, there seems to be no other source of information about this than Bell. His account, at any rate, became an accepted mathematical urban legend that has been repeated over and over again, often extending the three years of Bell to "twenty years of Sunday afternoons"…
Briefly stated, Cole relied on techniques such as introduced by Legendre and used existing tables of quadratic remainders… He discussed thoroughly the possible candidates of factors obtained with the help of this technique, together with some specific considerations for the case in point, and gradually focused on a reduced number of candidates which he tried one by one until he found the result. Cole was aware of Lucas’ announcement that $2^{67}-1$ and $2^{89}-1$ are composite... From all we know, the calculations he did, whether they took three years of Sunday afternoons or not, were done manually and without the aid of any mechanical device.
