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The famous anecdote of the 1903 announcement of the factorization of $2^{67}-1$ by Frank Nelson Cole has recently been discussed, for example in light of the announcement of another "twitter-sized" proof by Andrew Booker that $x^3+y^3+z^3=33$ has a solution.

Although there is already a fantastic breakdown of the story on HSM, and an even further review of the mathematics used by Cole on MathOverflow, I wonder if there's more evidence to be found to shed light on how much, if any, of the Cole story is true.

I'll take it as granted that Cole actually did factor $M_{67}$, because he has a paper to prove it.

The earliest source of the anecdote seems to be E.T. Bell's account from the "Queen of Mathematics" chapter in James Roy Newman's edited The World of Mathematics, Vol. 1.

Bell is silent on how long the multiplication would have taken. But there are other, secondary sources that expand on this. For example, Gridgemen states "Subsequently, in private, Cole said that those few minutes at the blackboard had cost him three years of Sundays." However, Wikipedia (March 19, 2019) states "Cole returned to his seat, not having uttered a word during the hour-long presentation."

Because as far as can be traced the story seems to originate from Bell, many have taken that to establish a prima facie case that the entirety of the story is not to be believed. But, is that harsh to Mr. Bell? A broken clock is right twice a day.

At least one question can be asked and answered:

Granting the necessary parts of the story, how long would it take for Cole to actually to do the multiplication on the blackboard?

Perhaps people would not sit through an hour of silent multiplication as Wikipedia suggests, but would sit through a few minutes, as Gridgemen suggests. My rough guess is that it's closer to ten minutes than an hour.

I will perform the experiment shortly.

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  • $\begingroup$ Three questions are answerable in some way? Doubtful. I think: Cole did not actually carry out calculations at the blackboard at the meeting. This story was invented by E. T. Bell. We also know of other stories in Men of Mathematics that were exaggerated or sensationalized. Reading The Search for E T Bell by Constance Reid, we find that Bell even fictionalized certain details of his own life. $\endgroup$
    – Gerald Edgar
    Mar 20, 2019 at 0:35
  • $\begingroup$ Possible duplicate of What's the famous story about a mathematician who gave a talk without saying a word? $\endgroup$
    – Conifold
    Mar 20, 2019 at 0:40
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    $\begingroup$ "... a broken clock...." when my clock breaks, it reads "88:88" :-) $\endgroup$
    – Carl Witthoft
    Mar 20, 2019 at 13:24
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    $\begingroup$ Based on the end of your post, that experiment is taking longer than an hour... $\endgroup$
    – KCd
    Mar 21, 2019 at 18:08
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    $\begingroup$ @MarkS okay, I have done the experiment of just multiplication of the two factors (not computing powers of 2 to find $2^{67}-1$ directly). I am not near a classroom, so I did it by hand on paper and used a phone's stopwatch to time myself. (Probably on a board it would have taken longer, since the calculation isn't all in your field of view in one place.) A few times I had to stop and recheck calculations. After 11 minutes and 7 seconds I was done and got a 21-digit number that looks like $147 \ldots 927$. This fits the first and last 3 digits of $2^{67}-1$, but 9 of 21 digits were wrong. $\endgroup$
    – KCd
    Mar 21, 2019 at 20:39

2 Answers 2

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Hand-Computation of $193707721 \times 761838257287$

This is a $9$-digit number times a $12$-digit number, and thus would require $9 \times 12 = 108$ digit-by-digit products, some of which require carrying, and then adding $12$ columns each with $9$ digits (or adding $9$ columns each with $12$ digits). Assuming each of the $108$ multiplications takes $3$ to $5$ seconds (this includes writing things down on the blackboard and sometimes carrying) and that adding the columns takes about $2$ minutes, we get roughly $7.5$ to $11$ minutes.

However, since there will be three $7$'s in the "bottom number" (no matter which number you use for the "bottom number"), you'll get close to a minute savings because for the second and third times you multiply the "top number" by $7,$ you simply copy down what you did the first time (with an appropriate number of $0$'s to the right). Also, when multiplying the "top number" by $1$ you just copy down the "top number". Finally, Cole had certainly done this several times previously, if only to check his work, so much of the computation would have been familiar to him.

Thus, I would bet the multiplication was done on the blackboard in roughly $5$ minutes $(2$ to $3$ minutes if he was alone, writing on paper), especially given that his skills in carrying out such computations were nowhere near as atrophied as is the case for most everyone today (due to calculators).

Hand-Computation of $2^{67} - 1$

My guess is that he began with a power of $2$ that everyone knew, such as $2^{10} = 1024,$ and worked up from that. Probably quickest would be to square this, then multiply the result by $1024$ to obtain $2^{30},$ then square this last number to get $2^{60},$ and finally multiply $2^{60}$ by $128.$ My guess is this would take him about $6$ to $10$ minutes. I used an on-line calculator and getting to $2^{20}$ took $30$ seconds (working quickly, but not trying to race the clock), but of course with three of the digits being $0$ and $1$ and $2,$ little work was needed.

Conclusion

My guess is that the entire process could probably be done in a relatively leisurely but steady manner within $20$ minutes.

For what it's worth, I used to do long multiplications like this from time to time before anyone in my school had a calculator (roughly when I was age $11$ to $16).$ For example, when checking results I saw in books or articles (such as Martin Gardner's "Mathematical Games" articles I saw in library copies of Scientific American or in a library book of his articles, and similarly for some of Isaac Asimov's well known essays), or working out large powers of $2$ and $3,$ etc. Anyway, I have a rough idea of how much work this would involve, and while it might appear daunting to someone today, it's not really all that much. (What I did a year ago here, now that was daunting.) It's just not something one would do in a talk, even back then. Of course, what "cost him three years of Sundays" were the computations needed to discover the factors to begin with.

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    $\begingroup$ Also count the time to compute $2^{67}-1$ by hand. $\endgroup$
    – KCd
    Mar 20, 2019 at 9:15
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    $\begingroup$ @KCd: I didn't realize that the power of $2$ was included until I saw your comment just now and went back to look at the cited web pages a bit more closely. I originally assumed that since powers of $2$ were something one would just look up in a math handbook, he would not have bothered with that aspect. $\endgroup$
    – Dave L Renfro
    Mar 20, 2019 at 10:31
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    $\begingroup$ It makes Bell's story about Cole even more ridiculous. Admittedly someone doing 2-powers by hand could just start off at $2^{10} = 1024$, well $2^{11} = 2048$, well $2^{12} = 4096$ say. Whether or not textbooks had a table of 2-powers, a lecture in the early 20th century couldn't assume people knew those powers anywhere close to the 67-th power or that they could look them up from a device in their pocket. $\endgroup$
    – KCd
    Mar 20, 2019 at 14:01
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    $\begingroup$ @KCd: References having tables of powers of $2$ would have been well known to Cole, and probably many could be found in the mathematical tables section of most any reasonably decent library at that time. For example, see these various 19th century references to such tables. Thus, I find the part about computing $2^{67} - 1$ a bit more likely to be made up than the part about multiplying the two factors of $2^{67} -1.$ $\endgroup$
    – Dave L Renfro
    Mar 20, 2019 at 16:08
  • $\begingroup$ You are correct about the time to multiply the two numbers: it took me just over 11 minutes. And it turns out I miscalculated a bunch of digits in the final answer. See one of my comments to the posted question. $\endgroup$
    – KCd
    Mar 21, 2019 at 20:41
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As it happens, the same question was covered in a submission to SIGBOVIK 2019 dated 13 March 2019 and published as pages 225 to 229 of the proceedings -- four methods were examined (the author describes long multiplication as lattice multiplication but worse and thus it was not tested) with the following results:

  • Lattice multiplication: 11 minutes
  • Double and halve: 21 minutes, gave incorrect answer
  • Quarter-square multiplication: Not tested, but given lookup tables up to 200,000 the author estimates 12 lookups would be required, with each lookup taking 30 seconds to a minute
  • Karatsuba multiplication: 17 minutes, acknowledged as anachronistic

The tests were performed on paper; the overhead of performing multiplications on chalkboard was not examined.

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