... Intensively cultivated for its own fascinations by hundreds of mathematicians of very different tastes, [number theory] has developed into an ever-growing expanse of loosely coordinated results with fewer general methods than any other major division of modern mathematics.

From all this heterogeneous miscellany we [find] only three topics in which there is some coherence of method and an approach to completeness in certain details. The rest is largely a wilderness of dislocated facts offering a strange and disconcerting contrast to the modernized generality of algebra, geometry, and analysis. Much of it is hopelessly archaic in both aim and results. [Number theory] appears to be the one remaining major department of mathematics where generalizing a problem makes it harder instead of easier. Consequently it has attracted fewer merely able young mathematicians than any other.

...

A problem in [number theory] is said to have been solved when a process is described whereby the required information is obtainable by a finite number of non-tentative operations. Time certainly is not the essence of the contract between the ... integers and the human intellect. The problem of resolving a number into its prime factors is solvable; yet the finite number of operations at present required for a number of a few thousand digits might consume more ages than our race is likely to have at its disposal.

The problem of finding the prime factors of a number must strike an amateur as a natural one. To say that it has been solved in any respect that would satisfy common sense is a flattering exaggeration, and the like is true of many other arithmetical problems that seem natural to the inexperienced. The professional ignores these natural problems in favor of others which he or his predecessors have constructed, and for which he may hope to find at least partial solutions. Complete solutions, even of manufactured problems, are comparatively rare .... Compared with what we should like to know in each of several directions, such progress as has been made is almost negligible. Yet all the resources of algebra and analysis have been hurled into the assault on this most elementary of all divisions of mathematics.