G H Hardy was President of Section A of the British Association for the Advancement of Science in 1922. The Association met in Hull in September and Hardy addressed Section A - Mathematical and Physical Sciences. Below is the second part of his lecture. To read the first part of Hardy's lecture, follow the link: British Association 1922, Part 1

G H Hardy, the President, continued his Address:-

3. I will next raise the question whether the number 2¹³⁷ - 1 is prime. I said that I would include one question which did not interest me particularly, and I should like to explain to you the kind of reasons which damp down my interest in this one. I do not know the answer, and I do not care greatly what it is.

The problem belongs to the theory of the so-called 'perfect' numbers, which has exercised mathematicians since the times of the Greeks. A number is perfect if, like 6 or 28, it is the sum of all its divisors, unity included. Euclid proved that the number

2^m(2^m+1 - 1)

is perfect if the second factor is prime; and Euler, 2,000 years later, that all even perfect numbers are, of Euclid's form. It is still unknown whether a perfect number can be odd.

It would obviously be most interesting to know generally in what circumstances a number 2ⁿ - 1 is prime. It is plain that this can only be so if n itself is prime, as otherwise the number has obvious factors; and the 137 of my question happens to be the least value of n for which the answer is still in doubt. You may perhaps be surprised that a question apparently so fascinating should fail to arouse me more.

It. was asserted by Mersenne in 1644 that that only values of n, up to 257, for which 2ⁿ - 1 is prime are

2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257;

and an enormous amount of labour has been expended on attempts to verify this assertion. There are no simple general tests by which the primality of a number chosen at random can be determined, and the amount of computation required in any particular case may be quite appalling. It has, however, been imagined that Mersenne perhaps knew something which later mathematicians have failed to rediscover. The idea is a little fantastic, but there is no doubt that, so long as the possibility remained, arithmeticians were justified in their determination to ascertain the facts at all costs. 'The riddle as to how Mersenne's numbers were discovered remains unsolved,' wrote Mr Rouse Ball in 1891. Mersenne, he observes, was a good mathematician, but not an Euler or a Gauss, and he inclines to attribute the discovery to the exceptional genius of Fermat, the only mathematician of the age whom anyone could suspect of being hundreds of years ahead of his time.

These speculations appear extremely fanciful, for the bubble has at last been pricked. It seems now that Mersenne's assertion, so far from hiding unplumbed depths of mathematical profundity, was a conjecture based on inadequate empirical evidence, and a rather unhappy one at that. It is now known that there are at least four numbers about which Mersenne is definitely wrong; he should have included at any rate 61, 89, and 107, and he should have left out 67. The, mistake as regards 61 and 67 was discovered as long ago as 1886, but could be explained with some plausibility, so long as it stood alone, as a merely clerical error. But when Mr R E Powers, in 1911 and 1914, proved that Mersenne was also wrong about 89 and 107, this line of defence, collapsed, and it ceased to be possible to take Mersenne's assertion seriously.

The facts may be summed up as follows. Mersenne makes fifty-five assertions, for the fifty-five primes from 2 to 257. Of these assertions forty are true, four false, and eleven still doubtful. Not a bad result, you may think; but there is more to be said. Of the forty correct assertions many, half at least, are trivial, either because the numbers in question are comparatively small, or because, they possess quite small and easily detected divisors. The test cases are those in which Mersenne asserts the numbers in question to be prime, there are only four of these cases which are difficult and in which the truth is known; and in these Mersenne is wrong in every case but one.

It seems to me, then, that we must regard Mersenne's assertion as exploded; and for my part it interests me no longer. If he is wrong about 89 and 107, I do not care greatly whether he is wrong about 137 as well or not, and I should regard the computations necessary to decide as very largely wasted. There are so many much more profitable calculations which a computer could undertake.

I hope that you will not infer that I regard the problem of perfect numbers, as uninteresting in itself; that would be very far from the truth. There are at least two intensely interesting problems. The first is the old problem, which so many mathematicians have failed to solve, whether a perfect number can be odd. The second is whether the number of perfect numbers is infinite or not. If we assume that all perfect numbers are infinite, we can state this problem in a still more arresting form. Are there infinitely many primes of the form 2ⁿ - 1? I find it hard to imagine a problem more fascinating or more terribly difficult than that. It is plain, though, that this is a question which computation can never decide, and it is very unlikely that it can ever give us any data of serious value. And the problem itself really belongs to a different chapter of the theory, to which I should like next to direct your attention.

4. Are there infinitely many primes of the form n² + 1? Let me first remind you of some well-known facts in regard to the distribution of primes.

There are infinitely many primes; their density decreases as the numbers increase, and tends to zero when the numbers tend to infinity. More accurately, the number of primes less than x is, to a first approximation,

x / log x

The chance that a large number n, selected at random, should be prime is, we may say, about 1/log n. Still more precisely, the 'logarithm-integral'

Li x = dt/log t (where the integral is from 2 to x)

gives a very good approximation to the number of primes. This number differs from Li x by a function of x which oscillates continually, as Mr Littlewood, in defiance of all empirical evidence to the contrary, has shown, between positive and negative values, and is sometimes large, of the order of magnitude √x or thereabouts, but always small in comparison with the logarithm-integral itself.

Except for one lacuna, which I must pass over in silence now, this problem of the general distribution of primes, the first and central problem of the theory, is in all essentials solved. But a variety of most exciting problems remain as to the distribution of primes among numbers of special forms. The first and simplest of these is that of the arithmetical progressions: How are the primes distributed among all possible arithmetical progressions an + b? We may leave out of account the case in which a and b have a common factor; this case is trivial, since an + b is then obviously not prime.

The first step towards a solution was made by Dirichlet, who proved for the first time, in 1837, that any such arithmetical progression contains an infinity of primes. It has since been shown that the primes are, to a first approximation at any rate, distributed evenly among all the arithmetical progressions. When we pursue the analysis further differences appear; there are on the average, for example, more primes 4n + 3 than primes 4n + 1, though it is not true, as the evidence of statistics has led some mathematicians to conclude too hastily, that there is always an excess to whatever point the enumeration is carried.

The problem of the arithmetical progressions, then, may also be regarded as solved; and the same is true of the problem of the primes of a given quadratic form, say am² + 2bmn + cn², homogeneous in the two variables m and n. To take, for instance, the simplest and most striking case, there is the natural and obvious number of primes m² + n². A prime is of this form, as I have mentioned already, if and only if it is of the form 4k + 1. The quadratic problem reduces here to a particular case of the problem of the arithmetical progression.

When we pass to cubic forms, or forms of higher degree, we come to the region of the unknown. This, however, is not the field of inquiry which I wish now to commend to your attention. The quadratic forms of which I have spoken are forms in two independent variables m and n; the form n² + 1 of my question is a non-homogeneous form in a single variable n, the simplest case of the general form an² + 2bn + c. It is clear that one may ask the same question for forms of any degree: Are there, for example, infinitely many primes n² + 2 or n⁴ + 1? I do not choose n³ + 1, naturally, because of the obvious factor n + 1.

This problem is one in which computation can still play an important part. You will remember that I stated the same problem for perfect numbers. There a computer is helpless. For the numbers 2ⁿ - 1, which dominate the theory, increase with quite unmanageable rapidity, and the data collected by the computers appear, so far as one can judge, to be almost devoid of value. Here the data are ample, and, though the question is still unanswered, there is really strong statistical evidence for supposing a particular answer to be true. It seems that the answer is affirmative, and that there is a definite approximate formula for the number of primes in question. This formula is

1/2 Li √x (1 + 1/3 ) (1 - 1/5) (1 + 1/7) (1 + 1/11 ) . . . .

where the product extends over all primes p, and the positive sign is chosen when p is of the form 4n + 3. Dr A E Western has submitted this formula to a most exhaustive numerical check. It so happens that Colonel Cunningham some years ago computed a table of primes n up to the value 15,000 of n, a limit altogether beyond the range of the standard factor tables, and Cunningham's table has made practicable an unusually comprehensive test. The actual number of primes is 1199, while the number predicted is 1219. The error, less than 1 in 50, is much less than one could reasonably expect. The formula stands its test triumphantly, but I should be deluding you if I pretended to see any immediate prospect of an accurate proof.

5. The last problem I shall state to you is this: Are there infinitely many prime-pairs p, p + 2? One may put the problem more generally: Does any group of primes, with assigned and possible differences, recur indefinitely, and what is the law of its recurrence?

I must first explain what I mean by a 'possible ' group of primes. It is possible that p and p + 2 should both be prime, like 3, 5, or 101, 103. It is not possible (unless p is 3) that p, p + 2 and p + 4 should all be prime!, for one of them must be a multiple of 3: but p, p + 2, p + 6 or p, p + 4, p + 6 are possible triplets of primes. Similarly

p, p + 2, p + 6, p + 8, p + 12

can all be prime, so far as any elementary test of divisibility shows, and in fact 5, 7, 11, 13 and 17 satisfy the conditions. It is easy to define precisely what we understand by a 'possible' group. We mean a group whose differences, like 0, 2, 6, have at least one missing residue to every possible modulus. The 'impossible' group 0, 2, 4 does not satisfy the condition, for the remainders after division by 3 are 0, 2, 1, a complete set of residues to modulus 3. There is no difficulty in specifying possible groups of any length we please.

We define in this manner, then, a 'possible' group of primes, and we put the questions: Do all possible groups of primes actually occur, do they recur indefinitely often, and how often on the average do they recur? And here again it would seem that the answers are affirmative, that all possible groups occur, and continue to occur for ever, and with a frequency whose law can be assigned. The order of magnitude of the number of prime-pairs, p, p + 2, or p, p + 4, or p, p + 6, both of whose members are less than a large number x, is, it appears,

x/(log x)²

The order of magnitude of the corresponding number of triplets, of any possible type, is

x/(log x)³

and so on generally. Further, we can assign the relative frequencies of pairs or triplets of different types; there are, for example, about twice as many pairs whose difference is 6 as pairs whose difference is 2. All these results have been tested by actual enumeration from the factor tables of the first million numbers; and a physicist would probably regard them as proved, though we of course know very well that they are not.

There is a great deal of mathematics the purport of which is quite impossible for any amateur to grasp, and which, however beautiful and important it may be, must always remain the possession of a narrow circle of experts. It is the peculiarity of the theory of numbers that much of it could be published broadcast, and would win new readers for the Daily Mail. The positive integers do not lie, like the logical foundations of mathematics, in the hardly visible distance, nor in the uncomfortably tangled foreground, like the immediate data of the physical world, but at a decent middle distance, where the outlines are clear and yet some element of mystery remains. There is no one so blind that he does not see them, and no one so sharp-sighted that his vision does not fail; they stand there a continual and inevitable challenge to the curiosity of every healthy mind. I have merely directed your attention for a moment to a few of the less immediately conspicuous features of the landscape, in the hope that I might sharpen your curiosity a little, and that some of you perhaps might feel tempted to walk a little nearer and take a rather closer view.

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