22 



REPORT 1877. 



Meissel has given a list of errata in Gauss's first million ; these I have confirmed ; 

 and I have given an errata list for the second and third millions. 



I hope to give hereafter the results of the enumeration in more detail, adding 

 the theoretical values for each group of 10,000 or 50,000 given by the li x formula, 

 by Legendre's formula, and by Riemann's formula i ' ilathematische Werke,' 1876, 

 pp. 136-144). 



Whenever there is no prime in a century, there must be a succession of at least 

 100 consecutive composite numbers ; and whenever there is only one prime, there 

 must be a succession of at least 50 ; but, of course, a 2-prime century or a 3-prime 

 century need not indicate a long sequence. In order to find long successions of 

 composite numbers I had all the instances in which there were 0, 1, 2, or 3 primes 

 in a century looked out in the tables, and the cases noted down in which the 

 sequence was fifty or upwards. These lists, for the six millions, were exhibited to 

 the Meeting at Plymouth. I only here give, in the next Table, some of the longest 

 sequences found in this manner, viz. those which exceed 110 in the first three 

 millions, and those which exceed 120 in the other three millions. 



Thus the 111 numbers between 370,261, and 370,373 are composite, and so on, 



