### Lecture sans paroles: the factors of M67

In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause.

Cole, an American mathematician born in 1861, was educated at Harvard. He lectured there and later at the University of Michigan and at Columbia University, New York. He spent 1883-85 in Germany, studying with the great mathematician Felix Klein in Leipzig and also visiting Göttingen. Cole completed his doctoral thesis, A Contribution to the Theory of the General Equation of the Sixth Degree, in America but under the guidance of Klein.

Cole was active in the American Mathematical Society, being Secretary from 1896 until 1920 and Editor-in-Chief of the AMS Bulletin. His main research interests were in algebra and number theory and he established the Cole Prizes, awarded by AMS for outstanding contributions to these two fields.

The Lecture sans Paroles

One of Cole’s doctoral students was Eric Temple Bell, remembered for his book Men of Mathematics. Bell relates the story of an extraordinary lecture to the American Mathematical Society on October 31st 1903, in which Cole identified the factors of the Mersenne number 267 – 1, or M67. The French mathematician Edouard Lucas had shown in 1876 that this number is composite, but he had not identified its prime factors.

Felix Klein.                                       Frank Cole.                                       Eric Temple Bell.

When the chairman called upon him, Cole – a man of few words – walked to the blackboard and calculated the value of M67 by computing increasing powers of 2 and finally subtracting 1. The result was the 21-digit number

147 573 952 589 676 412 927 .

Cole then moved to the other side of the blackboard and wrote the product of the 9-digit number 193 707 721 and the 12-digit number 761 838 257 287. After a tedious calculation, he obtained the result

193 707 721 x 761 838 257 287 = 147 573 952 589 676 412 927

which was identical to the Mersenne number M67. He then returned to his seat without having spoken a single word. For the first time on record, the audience responded with a standing ovation.

Cole had found the prime factors of the Mersenne number M67, a monumental computational task. He gave an account of his method in a paper, On the Factoring of Large Numbers, published later in 1903 in the AMS Bulletin.

It is worth mentioning that when the command FactorInteger[2^67-1] is entered into the programme Mathematica, the prime factors appear almost instantly. Likewise, typing factor(2^67-1) in SageMathCloud gives the factors within a split second.

Sources

Bell, Eric T., 1952: Mathematics, Queen and Servant of Science. G. Bell & Sons (See page 228).

Cole, F. N., 1903: On the factoring of large numbers, Bull. Amer. Math. Soc. 10, 134–137.