Died: 17 January 1675 in Paris, France

**Frenicle de Bessy** was an excellent amateur mathematician who held an official position as counsellor at the Court of Monnais in Paris.

He corresponded with Descartes, Fermat, Huygens and Mersenne. Most of the correspondence between these men and Frenicle de Bessy was on number theory but not exclusively so. He does comment on applied mathematical problems such as the trajectory of a body which falls from a starting position with an initial horizontal component. In a letter which he wrote at Dover in England to Mersenne on 7 June 1634, Frenicle describes an experiment to study the trajectory of a body released from the top of the mast of a moving ship. The data which he presents in the letter is quite accurate. Again on a more applied mathematical topic, Frenicle wrote an article which makes comments on Galileo's *Dialogue.*

Frenicle de Bessy is best known, however, for his contributions to number theory. He solved many of the problems posed by Fermat introducing new ideas and posing further questions. We shall look at some of the problems which were typical of those he worked on.

On 3 January 1657 Fermat made a challenge to the mathematicians of Europe and England. He posed two problems (in words rather than using notation as we shall do) involving *S*(*n*), the sum of the proper divisors of *n*:

1. Find a cube

nsuch thatn+S(n) is a square.2. Find a square

nsuch thatn+S(n) is a cube.

We know that Frenicle found four solutions to the first of these problems on the day that he was given the problem, and found another six solutions the next day. He gave solutions to both problems in *Solutio duorm problematum ...* (1657). In this work he posed some problems of his own, including the following:

Find an integer

nsuch thatS(n) = 5n, andS(5n) = 25n.Find an integer

nsuch thatS(n) = 7n, andS(7n) = 49n.Find

nsuch thatn^{3}- (n-1)^{3}is a cube.

Frenicle solved other problems posed by Fermat. For example he showed that if a right angled triangle has sides integers *a*, *b*, *c* then its area *bc*/2 can never be a square. He also showed that the area of a right angled triangle is never twice a square.

Frenicle de Bessy also worked on magic squares and published *Des quarrez ou tables magiques*. He was elected to the Académie Royale des Sciences in 1666.

**Article by:** *J J O'Connor* and *E F Robertson*

**July 2000**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Frenicle_de_Bessy.html]