NA TURE 



[September 16, 1922 



Still more precisely, the " logarithm-integral " 

 r x it 



Li x = 



.log/ 



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 

 interesting 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 

 4« + 3 than primes 4% + 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 am- + ibmn + en 2 , homogeneous in the two variables 

 //; and n. To take, for example, the simplest and 

 most striking case, there is the natural and obvious 

 number of primes m 2 + n 2 . 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 

 progressions. 



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 n 2 +i of my question is 

 a non-homogeneous form in a single variable n, the 

 simplest case of the general form an 2 + 2bn + c. It is 

 clear that one may ask the same question for forms of 

 any degree : are there, for example, infinitely many 

 primes rc 3 + 2 or » 4 +i? I do not choose « 3 +i, 

 naturally, because of the obvious factor n + i. 



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 2 n - i. which 

 dominate the theory, increase with unmanageable 



NO. 2759, VOL. I io] 



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 



where the product extends over all primes p, and the 

 positive sign is chosen when p is of the form 4W + 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 « 2 +i 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 1 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 1 



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 + &,p + 12 



can all be prime, so far as any elementary test of 

 divisibility shows, and in fact 5, 7, n, 13 and 17 

 satisfy the conditions. It is easy to define precisely 

 what we understand by a " possible " group. We 

 mean a group the differences in which, like o, 2, 6, 

 have at least one missing residue to every possible 

 modulus. The "impossible" group o, 2, 4 does not 

 satisfy the condition, for the remainders after division 

 by 3 are o, 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 in- 

 definitely often, and how often on the average do they 

 recur ? 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 the 

 law ni which 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 members of which are less than a large 

 number x, is, it appears, 



(loglvF 



