luith Goldhach's Theorem 239 



§ 2. For the numerical data used we are indebted to two 

 different sources. The most complete numerical results are con- 

 tained in the tables compiled and published* by R. Haussner, which 

 give the values of v(n) for all values of n not exceeding 5000. 

 Tables extending up to 1000 and 2000 had been calculated earlier 

 by G. Cantor and V. Aubry. Further data, less systematic, indeed, 

 than those of Haussner, but extending to considerably larger values 

 of n, were given by L. Ripertf in a number of short papers in 

 V Intermediaire des mathe'inaticiens. 



The values given for v()i) in the accompanying table differ, in 

 several respects, from those given by Haussner or Ripert. In the 

 first place, 7n + m and m + m are here counted as different decom- 

 positions, whereas the above two writers regard them as identical ; 

 secondly we do not (as do Haussner and Ripert) regard 1 as a 

 prime ; and thirdly we increase the values of v (n) obtained from 

 their tables by addition of the number of ways in which n may be 

 expressed as the sum of two powers of primes, i.e. the number of 

 ways in which 



n=p'^ + (f', 

 where j) and q are primes, and either a or b is greater than unity. 

 The last two modifications make, of course, no difference to the 

 asymptotic formula, but it seems natural to make them when the 

 genesis of the formula (1) or (5) is considered. 



As regards the choice and arrangement of the numbers n in the 

 table, the smaller numbers — i.e. the numbers not exceeding 5000 

 — are intended to be " typical " ; that is, they are specially selected 

 numbers, taken in groups so as best to test or illustrate the accuracy 

 of formula (1). Thus, for example, if the formula in question is true, 

 a multiple of 6 may be expected, in general, to allow of an unusually 

 large number of decompositions :|:. On the other hand a power of 2 

 may be expected to allow of an unusually small number. The 

 numbers below 5000 have therefore been selected in groups of four 

 or five, all the numbers of each group being as nearly equal as 

 possible ; and each group of numbers contains, in general, one 

 highly composite number (i.e. 2.3.5.7,11....), one power of 2, 

 and one number which is the product of 2 and a prime. 



For values of n exceeding 5000, such choice of" typical " numbers 

 was, unfortunately, impossible without a large amount of fresh 

 calculation. Ripert, indeed, selected his numbers according to a 

 system, and they, too, occur, in general, in gToujJs of approximately 

 equal magnitude; but he selected them with different objects, so 

 that his numbers are, from our point of view, neither " typical " nor 

 arbitrary. 



* Nova Acta tier Akad. der Natur/orscher (Halle), vol. 72 (1897), pp. 5-214. 

 t See, for example, vol. 10 (1903), pp. 76-77, 16(3-167. 



X It was first pointed out by Cantor, on the evidence of his numerical results 

 previously mentioned, that this is actually so. 



VOL. XIX. PART V. 17 



