Carl Ferdinand Degen (1 November 1766 – 8 April 1825) was a Danish mathematician. His most important contributions were within number theory and he advised the young, aspiring Norwegian mathematician Niels Henrik Abel in a decisive way. Degen has received much of the credit for the introduction of more modern and advanced mathematics in the Danish-Norwegian school system.
He was born in Braunschweig in Germany, but the family moved to Copenhagen in 1771 when his father Johan Philip Degen got a position in the Royal Danish Orchestra. As a musician he had a low salary, but his son Carl Ferdinand received a fellowship so that he could go to school in Helsingør. He graduated from there in 1783 and continued at the University of Copenhagen. Instead of following the normal path of studies, the young Degen followed his own interests and read classical languages, philosophy, natural sciences and in particular mathematics. When the university in 1792 for the first time announced a prize essay contest in several different fields with an award of 40 riksdaler in each, Degen won the prize both in theology and in mathematics. He was fluent in Latin, Greek and Hebrew, was well-acquainted with Romance and Germanic languages and could read Russian and Polish. In this period he was tutor in mathematics for the young prince who later became king Christian VIII of Denmark. In 1798 Degen was made a Doctor of Philosophy based on a thesis on Kant's philosophy and was elected to the Royal Danish Academy of Sciences and Letters in 1800.
In 1802 Degen got his first academic position as head teacher in mathematics and physics at the Odense cathedral school. After a few years there he was appointed rector at the corresponding school in Viborg. There he remained until 1814 when he became professor in mathematics at the University of Copenhagen. Although his lectures were not so well organized, he was loved by his students and he infused the courses with new and more modern mathematics. At the same time he pursued his own research and published results in many different directions. All this made him the most esteemed mathematician in Scandinavia at that time.
When Niels Henrik Abel as a student visited Degen in Copenhagen, he described him as very kind, but a little strange, with a large, private library. Degen remained there until his death in 1825. For that reason he did not live to see the great fame the young Abel shortly afterwards obtained from his discovery of elliptic functions which Degen had encouraged. He is buried on the Assistens Kirkegård at Nørrebro in Copenhagen.
Mathematical contributions
Degen worked in many branches of what was then modern mathematics. Most of his contributions had to do with problems within number theory, but he also wrote papers on geometry and mechanics.
The Pell equation
In 1817 Degen got printed his large work on the fundamental solutions (x, y) of Pell's equation x2 – ny2 = 1 where n is a positive integer. Euler had earlier shown that these could be systematically calculated with the use of continued fractions. Degen used this method and presented integer solutions for all The same calculations also gave approximate, but very accurate rational results for the square root of n. In addition, he also found solutions of the adjoint equation with −1 on the right hand side for the n-values when they existed. These tables of numerical results became in the following years a standard reference for the Pell equation.
The eight-square identity
While his work on the Pell equation can be considered a continuation of previous contributions made by Euler, Lagrange and Legendre to this problem, Degen's discovery of the eight-square identity was his most important and original discovery. Most probably it resulted from his attempts to generalize the Pell equation.
The two-square identity
had been known from the times of Diophantus. At the end of the 17th century it explained why the norm of the product of two complex numbers equals the product of their norms. Around the same time Euler showed that there is also a similar four-square identity. Later it turned out to be related to the norm of quaternions discovered by William Rowan Hamilton. In 1818 Degen presented to the Academy of Sciences in St. Petersburg where Euler had worked, his eight-square identity of exactly the same structure as the two previous identities. The following year he was elected as a «corresponding member» to the same academic society.