As we have already noticed, Fermat’s connection to other mathematicians was never an official one, but only consisted of  letters and seldom personal meetings. He never published anything under his name though he produced up to 3000 mathematical papers during his life, and he also disliked to announce proofs of theorems and conjectures he made. He enjoyed very much being the magician who enchanted his audience by performing miracoulous tricks of which only he knew the entire secrets.

But a mathematician cannot work on his own, at least not all the time. He needs the correspondence and inspiration from others, and so did Fermat. He was in very close friendship with Beaugrand,  a student of Viète and a mathematician and astronomer. Fermat and Beaugrand both shared great respect and admiration for Viète’s work.  Fermat frequently sent his new discoveries to Beaugrand to hear his opinion, even though Beaugrand could not reach his genius.

Fermat also became good friends with Carcavi, who was in correspondence with other great mathematicians like Huygens, Pascal, Galileo and Descartes. In 1650 Fermat sent Carcavi a treatise which contained the first known method of elimination. He wanted it published, but unfortunately Carcavi failed in finding interested publishers.

In 1636 Fermat learned to know Mersenne through Carcavi, who told Mersenne in details about Fermat’s genius. Mersenne wrote to Fermat and asked him a few questions on his new works and Fermat wrote back, telling him a lot more about his discoveries than Mersenne asked for. Thus a long lasting scientific correspondence started and marked a new era in Fermat’s mathematical career since Mersenne and his group helped him in circulating ideas and  arousing more interests from the academic society in Paris. Mersenne himself was very interested in number theory, which was quite rare for that time, and he investigated prime numbers and their general representation. Mersenne’s most famous work was on numbers of the form 2^p – 1,  (p prime).

He stated that     2^p – 1 is prime     p prime.

In 1644 Mersenne claimed that  2^p – 1 is prime if p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 but composite for the other 44 primes smaller than 257.  He was not entirely right, as was proved later, but he encouraged other mathematicians in investigating these numbers thereby contributing to number theory. His observations are also of use in finding large primes today.

As Fermat was a member of Mersenne’s mathematical group, so was Étienne Pascal, Blaise Pascal’s father.  Pascal heard of Fermat’s enormous talent from his father and so, when he could not find a solution to some problems on gaming and chances which built later the basis for probability theory, he wrote to Fermat and asked for his advice. In 1654, the same year, Fermat wrote back and solved the problems, thus founding, together with Pascal, the theory of probability.  There are  five letters between them remaining today.  Fermat tried to bring the topic of their discussion to number theory, but unfortunately Pascal was not really interested, so he did not edit Fermat’s works on number theory and other fields, as Fermat had hoped, and later Fermat put the idea of publication aside again.
 
 

Unfortunately, Fermat did not maintain friendly connections with another great mathematician of his time, Descartes, one of the most famous philosopher who once said “Cogito Ergo Sum” – “I think, therefore I am”. Descartes’ most important works in mathematics are the three parts of his Discours de la méthod pour bien conduire sa raison et chercher la vérité dans les sciences. These three parts are La Dioptrique, La Méteores, and the most important one, La Géomètrie. La Dioptrique was a work on optics and didn't contain much new discoveries, La Méteores was about meteorology and was only important for its position as the first scientific investigation of the weather, though there were large numbers of errors which could have easily been corrected if Descartes would have taken the time to do some experiments to support his statements. Finally, La Géomètrie had the most impact on the scientific world since it showed for the first time the correlation between geometry and algebra, and since he did the first steps towards a theory of invariants.

Fermat became involved in a controversy with Descartes who was very displeased about Fermat’s careless way of commenting on his work La Dioptrique, consisting of observations of light and its refraction. Later, when Descartes had the opportunity to view Fermat’s work on maxima and minima, he got even more angry since he thought of Fermat’s work as a mere mockery of his own work, La Géomètrie, which he thought was one of his greatest works and a completely new way of geometrical calculation. Descartes attacked Fermat’s methods on maxima and minima. Because of his great popularity in the mathematical society,  his attack soon became so widely known that it was impossible for other leading mathematicians to stay neutral. Finally Roberval, Pascal and Desargues became involved, and Fermat’s reputation was badly damaged, though Descartes eventually had to admit that Fermat’s methods were true: “... seeing the last method that you use for finding tangents to curved lines, I can reply to it in no other way than to say that it is very good and that, if you had explained it in this manner at the outset, I would have not contradicted it at all.”