A.— MATHEMATICS AND PHYSICS. 23 



Are lliure, I'ui- example, inhnitely iiuiuy primes tv' + 2 or n' + l'.' 1 do 

 not choose n^ + 1, naturally, because of the obvious factor ii + l. 



This problem is one in which computation can still play an im- 

 portant part. You will remember that I stated the same problem for 

 perfect numbers. There a computer is helpless. For the numbers 

 2" - 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 



iV-(i+D0-5)0+7)(^+A 



where the j^roduct extends over all primes p, and the positive sign is 

 chosen when p is of the form 471+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- + 1 

 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 + Q 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 



