Waring wrote a number of papers in the "Philosophical Transactions of the Royal Society", dealing with the resolution of algebraic equations, number theory, series, approximation of roots, interpolation, the geometry of conic sections, and dynamics. The "Meditationes Algebraicae" (1770), where many of the results published in "Miscellanea Analytica" were reworked and expanded, was described by Joseph-Louis Lagrange as 'a work full of excellent researches'. In this work Waring published many theorems concerning the solution of algebraic equations which attracted the attention of continental mathematicians, but his best results are in number theory. Included in this work was the so-called Goldbach conjecture (every even integer is the sum of two primes), and also the following conjecture: every odd integer is a prime or the sum of three primes. Lagrange had proved that every positive integer is the sum of not more than four squares; Waring suggested that every positive integer is either a cube or the sum of not more than nine cubes. He also advanced the hypothesis that every positive integer is either a biquadrate (fourth power) or the sum of not more than nineteen biquadrates. These hypotheses form what is known as Waring's problem. He also published a theorem, due to his friend John Wilson, concerning prime numbers; it was later proven rigorously by Lagrange.