1. This note gives a brief account of a particular use of computers in number theory. It describes a part of some research done jointly by J. N. O'Brien and myself, which will appear in detail in the Transactions of the American Mathematical Society, where the full list of references will be available. I have subsequently, with the assistance of Atlas, extended these results in various ways, of which an account will be presented to the IBM Symposium on Utilisation of Computers in Mathematical Research at Blaricum, Holland, from 29th to 31st August 1966. In that account the concept of the computer as an aid to discovery is perhaps more apparent; on the other hand, it demands that the reader be familiar with the theory of modular forms. Accordingly, I have preferred for the present occasion to lose generality and, I hope, gain accessibility for the non-technical reader.
Cassells has observed that number theory is an experimental science. This remark is well exemplified by the work of Ramanujan (1887-1920) who combined a genius for empirical discovery with an extraordinary flair for (hand) computation. The particular aspect of his work we consider here is related to the number theoretic function p(n), defined as the number of unrestricted partitions of n, where partitions differing only in the arrangement of their parts are regarded as equivalent. Thus
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1,
so that p(5) = 7. It was proved by Euler that
(1)
and the product on the right-hand side of (1) regarded as a function of a complex variable x = e2π i τ is essentially a modular form. Ramanujan conjectured from an examination of tables (in this instance computed by Macmahon) the following properties.
If 24n-1 is divisible by 5α7β11γ, then p(n) is divisible by 5α7β11γ; where α, β and γ are any integers ≥ 1 (2)
He proved (2) for α ≥ 2, β ≥ 2, β ≥ 2. However, Gupta in 1935 found that p(243) is not divisible by 73, so that (2) fails for β = 3 since 24.243-1 is divisible by 73. We have here an immediate instance of the value of large and powerful computers; nowadays the natural course would be to test (2) a far as reasonably possible, and Atlas could test n ≤ 40,000 in about 3 minutes.
In fact the correct form of (2) is:
If 24n-1 is divisible by 5α7β11γ, then p(n) is divisible by 5α7β11γ; where b = (β + 1)/2 if β is odd and b = (β + 2) / 2 if β is even (3)
This was proved by Watson (1938) for 5 and 7 and Atkin (1966) for 11.
2. It is natural to search for properties involving primes q other than 5, 7 and 11. The cases q = 2 and 3 offer little prospect, essentially because 24n-1 cannot be divisible by 2 or 3. Also it is (by now) most unlikely that any results precisely analogous to (3) are available. However Newman (1957) proved the following result.
Let t1 = 13a+6, and define tm = tm(a) inductively by tm = 132 tm-l - 7.
Then p(tm) - 6p(tm-l) is divisible by 13. (4)
Newman now chooses the accidental value a = 6, t1 = 84, where p(84) is divisible by 13, and infers that p(tm(6)) is divisible by 13 for all m, thus exhibiting an infinity of n for which 13 divides p(n).
For the reader unfamiliar with number theory we fill in the argument a little. We know p(84) = 13n1, where n1 is an integer. By Newman's theorem p(14189) - 6. p(84) = 13n2 where n2 is an integer, and so p(14189) = 13(6n1 + n2) and is divisible by 13. The induction is now clear. The numbers tm(6) increase exponentially, and p(tm()) is of the order of ekm, so that no direct verification of Newman's result is practicable!
O'Brien and I proved a result analogous to (4) in which the conclusion refers to divisibility by a general power of 13. The application demands an accident similar to the fact that p(84) is divisible by 13; using Atlas we found such an accident (in fact, several) to give:
p(n) is divisible by 134 infinitely often. (5)
The proof of (5), then, requires first a general result proved by the normal processes of mathematics, and then a single numerical fact well beyond any hope of hand computation. It is a disputed point as to how far this constitutes proof in pure mathematics; the reader without a computer cannot verify the fact, and the reader with one may not feel justified in using valuable machine time to recompute it. My own view is that the theoretical distinction is valid, but that the mathematician who is compelled in practice to quote known results may attach the same reliability to a modern computer, properly used, as to published work which he does not personally verify. It would be invidious for me to give here relevant examples of results published, and subsequently quoted by other authors, which are in fact untrue; especially since a computer exposed their falsehood:
3. The computer's role as described in paragraph 2 was merely to find an accident relevant to a known general theorem. However, inspecting the tables produced in connection with (5), O'Brien and I observed the following:
If 24n-1 is divisible by 13m and p(n) is divisible by 13m, then so is p(N) where 24N -1 = r2 (24n-1) and r ≥ 5 is prime. (6)
This in its turn enabled us to make a more general conjecture as to a multiplicative property of p(n) in relation to divisibility by powers of 13, which I have recently proved.
The computer now appears more clearly as an agent of discovery; a final result can be stated and proved without recourse to any computed facts, but one which without the assistance of a large and fast machine might have remained unnoticed. It may be expected that, with the advent of immediate access through consoles, this kind of use will appear with increasing frequency. When there is no longer a delay between the immediate speculation inherent in this kind of research and the factual answer, number theorists will be in a position which even Ramanujan could not emulate. This is not, of course, to say that the human brain becomes unnecessary, as some popular accounts of computers would have us believe: Nor is it intended as any slight on the genius of Ramanujan; one may speculate as to what he would have achieved with the aid of computers had he lived 50 years later. I conjecture that whole areas of number theory would wear a different aspect, but this is a conjecture I cannot test on Atlas.
