Cambridge allows a Ph.D. under the "special regulations", which means on the basis of published papers. In 1977 I asked Professor Cassels, then head of the Pure Maths Department, about this. He encouraged me to apply and was one of the two examiners. I was awarded the degree in 1978 on the basis of the first six papers of my list (see my Number Theory Page). I had earlier pursued a Ph.D. (in number theory) in the conventional way - but had an unsatisfactory relationship with my elderly supervisor.
In 1968 I was working in a wooden hut grandly entitled "Mathematical Laboratory" in the vegetable garden at Taplow Court, the home of "Plessey Telecommunications Research" (formerly "British Telecommunications Research" - but nothing to do with today's BT). The amount of real research on the site was diminishing but in our hut there was an Elliott 4120 computer (Algol60 and paper tape), and interesting books (such as Knuth Vol. 2) and journals, especially Math. Comp., where computational number theory has always featured. My research was usually a spare time hobby. Doing research without a supervisor is difficult because you make bad choices of problem but if you can eventually write papers which are accepted you have perhaps achieved more. Relevant to mention that the main interest of the Math Lab was Telephone Traffic Theory, which involves random walks.
I left Taplow Court in 1986. In 1988, it became a centre for Buddhism!:
https://sgi-uk.org/Local-Community/SGI-Centres/Taplow-Court
More about the interesting history of Taplow Court is here:
https://en.wikipedia.org/wiki/Taplow_Court
Replies to some recent questions:
1. Did my interest in factoring come from the RSA method?
No. I was interested before. My papers are listed on my Number Theory Page:
My first paper about primality testing was published in 1971, and my first about factoring in 1974.
I first heard of the application of number theory to cryptography in the important paper of Diffie and Hellman(1976).
The RSA method was published by them in 1978.
2. What were my intentions in creating the first version of the Number Field Sieve(NFS)?
If you are once interested in factoring, you will probably continue to be. After having the good fortune to invent the rho method of factoring - paper 6(1975) - it seemed to me that if I was to make any further discoveries in number theory, this was the most likely area. I wondered if the ideas of that method could be applied to any other problem. And I found that the ideas could indeed be applied to the calculation of "discrete logarithms" (a problem very little known at that time). This is described in my paper 9(1978).
The invention of the (special) NFS came about as a result of my attempts to improve on the Quadratic Sieve of Pomerance(QS). I first thought of the degree 2 case of the NFS - for very special numbers. Then I tried to extend it to degree 3. At some point I noticed that what I was doing was close to the paper of Coppersmith, Odlyzko and Schroeppel(1986) which Odlyzko had sent me. As related in paper 14, I succeeded in repeating the factorisation of F7.
In 1996 Carl Pomerance wrote an excellent introduction to QS and NFS:
"A Tale of Two Sieves" (Notices of the AMS, December 1996, 1473-1485).
(last modified 13th October 2022.)