Her name was Sr. Mary Ferdinand Torline, CSJ. She might also be referred to as M. Ferdinand Torline, and the “Sr.” and “CSJ” might be omitted.
She wrote a master’s thesis on “The Practical Solution of the Cubic Binomial Congruence” at St. Louis University, in 1949.
She wrote her PhD thesis on “Waring’s Problem, Modulo P” at the same university, in 1955, with her advisor being John Dyer Elder.
She also wrote punch-card programs (I haven’t yet tracked down which language), which suggests that she remained active in her field after her PhD, since computers would have been quite rare in 1955.
More information is certainly available on the master’s thesis and the punch-card programs, as those are physically kept nearby. I was prompted onto this subject by the archivist in custody of said items being curious about them.
My first thought was to check Web of Science, but the last time I used that, I still had an institutional affiliation, and didn’t realize that it was subscription-only. My second thought was Google Scholar, but I couldn’t find much there.
In addition to knowing what other publications she wrote and who she was cited by, I’m also curious about what the implications of her work would be. On the introductory page of her master’s thesis, she acknowledged that the work had very little practical application, but much has changed in number theory, and in particular modular number theory, since 1949. Would that work be relevant to cryptographic hashes, for instance?
Torline, Sister Mary Ferdinand, C.S.J. “Waring’s Problem, Modulo p,”
Unpublished Ph.D. dissertation, Department of Mathematics, Saint Louis
University, 1955
The “unpublished” suggests that it might be hard to dig up more.
One other tidbit I found while researching her: In the entire decade 1950-1959, only 104 American women got PhDs in math… and of those 104, 12 were nuns.
I guess nuns don’t have much patience with people telling them what women ought and ought not to do.
Anyone look into where M. Anne Real went to college? The thought which comes to mind that she was a Master’s degree student of John Dyer Elder at Saint Louis University who was also the PhD advisor of Sr. Mary Ferdinand Torline–and M. Anne Real’s publication was a condensation of her master’s thesis.
Very curious. The mathematical genealogy search does not find her. I have searched for many on that site and never failed before.
Edited to add: Then I searched for John Dyer Elder and she came right up as one of his students with her full name and thesis title. She had no known students is all I can add.
I do intend to take a closer look at those punchcards, to see if I can figure out what the program was for (since it was probably something after her PhD). It can’t be more than a few hundred lines; I can suffer through that much Fortran.
Meanwhile, if we can’t find the publications or citations themselves, how about the other part, what her work might have been relevant for (even if it went unnoticed)? IIRC, raising numbers to some power, modulo some number, is one of the key operations in cryptography. Would cubic binomials be part of that?
Number theory is not my bag and there is not a lot I can add to the wiki article. Once you know that every number is the sum of 4 squares (not obvious, although I do know an elementary* proof) and every number is the sum of 9 cubes (I haven’t the faintest idea how to prove it), it seems natural to conjecture that every number is the sum of 16 4th powers. Until you run into the fact that 79 is not the sum of fewer than 19 4th powers (4 16s and 15 1s). I still think that there are only finitely many exceptions and that every sufficiently large number is the sum of 16 4th powers, but I could be wrong.
At any rate, Waring conjectured and Hilbert proved that for each natural number n, there is a number G(n) such that every number is the sum of G(n) nth powers. When n > 3, there is also a number g(n) < G(n) such that every sufficiently large number is the sum of g(n) nth powers.
*elementary does not always mean easy, only that no advanced concepts are used. In this case the most advanced idea used is modular arithmetic (arithmetic mod n).
