Some Famous Problems of the Theory of Numbers

SOME FAMOUS PROBLEMS OF THE THEORY OF NUMBERS. AND IN PARTICULAR WARING'S PROBLEMIt is expected that a professor who delivers an inaugural lecture should choose a subject of wider interest than those which he expounds to his ordinary classes. This custom is entirely reasonable; but it leaves a pure mathematician faced by a very awkward dilemma. There are subjects in which only what is trivial is easily and generally comprehensible. Pure mathematics, I am afraid, is one of them; indeed...
More Description

SOME FAMOUS PROBLEMS OF THE THEORY OF NUMBERS. AND IN PARTICULAR WARING'S PROBLEMIt is expected that a professor who delivers an inaugural lecture should choose a subject of wider interest than those which he expounds to his ordinary classes. This custom is entirely reasonable; but it leaves a pure mathematician faced by a very awkward dilemma. There are subjects in which only what is trivial is easily and generally comprehensible. Pure mathematics, I am afraid, is one of them; indeed it is more: it is perhaps the one subject in the world of which it is true, not only that it is genuinely difficult to understand, not only that no one is ashamed of inability to understand it, but even that most men are more ready to exaggerate than to dissemble their lack of understanding.There is one method of meeting such a situation which is sometimes adopted with considerable success. The lecturer may set out to justify his existence by enlarging upon the overwhelming importance, both to his University and to the community in general, of the particular studies on which he is engaged. He may point out how ridiculously inadequate is the recognition at present afforded to them; how urgent it is in the national interest that they should be largely and immediately re-endowed; and how immensely all of us would benefit were we to entrust him and his colleagues with a predominant voice in all questions of educational administration. I have observed friends of my own, promoted to chairs of various subjects in various Universities, addressing themselves to this task with an eloquence and courage which it would be impertinent in me to praise. For my own part, I trust that I am not lacking in respect either for my subject or myself. But, if I am asked to explain how, and why, the solution of the problems which occupy the best energies of my life is of importance in the general life of the community, I must decline the unequal contest: I have not the effrontery to develop a thesis so palpably untrue. I must leave it to the engineers and the chemists to expound, with justly prophetic fervor, the benefits conferred on civilization by gas-engines, oil, and explosives. I suppose that every mathematician is sometimes depressed, as certainly I often am myself, by this feeling of helplessness and futility. I do not profess to have any very satisfactory consolation to offer. It is possible that the life of a mathematician is one which no perfectly reasonable man would elect to live. There are, however, one or two reflections from which I have sometimes found it possible to extract a certain amount of comfort. In the first place, the study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation, and we have learnt that it is something to be able to say that at any rate we do no harm. Secondly, the scale of the universe is large, and, if we are wasting our time, the waste of the lives of a few university dons is no such overwhelming catastrophe. Thirdly, what we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. And, finally, the history of our subject does seem to show conclusively that it is no such mean study after all. The mathematicians of the past have not been neglected or despised; they have been rewarded in a manner, undiscriminating perhaps, but certainly not ungenerous. At all events we can claim that, if we are foolish in the object of our devotion, we are only in our small way aping the folly of a long line of famous men, and that, in these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.G. H. HARDY