Robinson received her PhD degree in 1948 under Alfred Tarski with a dissertation on "Definability and Decision Problems in Arithmetic". Her dissertation showed that the theory of the rational numbers was an undecidable problem, by demonstrating that elementary number theory could be defined in terms of the rationals. (Elementary number theory was already known to be undecidable by Gödel's first Incompleteness Theorem.) Here is an excerpt from her thesis:"This consequence of our discussion is interesting because of a result of Gödel which shows that the variety of relations between integers (and operations on integers) which are arithmetically definable in terms of addition and multiplication of integers is very great. For instance from Theorem 3.2 and Gödel's result, we can conclude that the relation which holds between three rationals A, B, and N if and only if N is a positive integer and A=BN is definable in the arithmetic of rationals." Hilbert's tenth problem Hilbert's tenth problem asks for an algorithm to determine whether a Diophantine equation has any solutions in integers. Robinson began exploring methods for this problem in 1948 while at the RAND Corporation. Her work regarding Diophantine representation for exponentiation and her method of using Pell's equation led to the J.R. hypothesis (named after Robinson) in 1950. Proving this hypothesis would be central in the eventual solution. Her research publications would lead to collaborations with Martin Davis, Hilary Putnam, and Yuri Matiyasevich. In 1950, Robinson first met Martin Davis, then an instructor at the University of Illinois at Urbana-Champaign, who was trying to show that all sets with listability property were Diophantine in contrast to Robinson's attempt to show that a few special sets — including prime numbers and the powers of 2 — were Diophantine. Robinson and Davis started collaborating in 1959 and were later joined by Hilary Putnam, they then showed that the solutions to a “Goldilocks” equation was key to Hilbert's tenth problem. In 1970, the problem was resolved in the negative; that is, they showed that no such algorithm can exist. Through the 1970s, Robinson continued working with Matiyasevich on one of their solution's corollaries, which she once stated that there is a constant N such that, given a Diophantine equation with any number of parameters and in any number of unknowns, one can effectively transform this equation into another with the same parameters but in only N unknowns such that both equations are