164 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



A. Tanturri 230 gave expressions for the number of partitions of n into 2, 

 3, 4 or 5 distinct parts, and recursion formulas. He 231 investigated the 

 number D n of partitions of n into powers of 2 and the number D(2 P , n) of 

 partitions of n into powers of 2 of which 2 P is the maximum. The first 

 function can be computed from the second. In the second paper occur 

 recursion formulas for the second function, and expressions for D(2 P , 2 p k) 

 and D(2 P , 2 p k + 2 p ~ l ) in terms of binomial coefficients. 



On the number of positive integral solutions of ax + by n, see papers 

 117-142a of Ch. II. Cesaro, Vol. I, p. 306, gave relations involving the 

 number of positive integral solutions of 1 + 2 2 + + v% v = n. 



Von Sterneck, Vol. I, p. 427, used partitions into elements formed from 

 the first s primes. 



230 Atti R. Accad. Sc. Torino, 52, 1916-7, 902-918. In Peano's symbolism, with a translation 



of most of the results. 



231 Ibid., Dec. 1, 1918. Continued in Atti R. Accad. Lincei, Rendiconti, 27, II, 1918, 399- 



403. In Peano's symbolism with partial translation. 



