His most important contributions were within number theory and he advised the young, aspiring Norwegian mathematician Niels Henrik Abel in a decisive way.
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.
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.
He was fluent in Latin, Greek and Hebrew, was well-acquainted with Romance and Germanic languages and could read Russian and Polish.
[2] 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.
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.
[3] 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.
He ended the letter with the wish that .... the time and efforts that Mr. Abel in my eyes spends on this rather sterile subject ought to be invested in a problem whose development will have the greatest consequences for Mathematical Analysis and its applications to practical investigations.
A serious investigator with suitable qualifications for research of this kind would by no means be restricted to the many strange and beautiful properties of these most remarkable functions, but could discover a Strait of Magellan leading into the wide expanses of a vast Analytic Ocean.This would soon turn out to be a very prophetic piece of advice.
In a letter to his friend and former teacher Bernt Michael Holmboe in Oslo he wrote that he had constructed elliptic functions by inverting the corresponding integrals.
[7] Although this discovery marks the beginning of a new and very important branch of modern mathematics, Abel waited with the publication of his results.