356 SECTIONAL TRANSACTIONS.—A*. 
discussion of the properties of certain specially constructed linear forms of 
the general type 
L(x) = P(x) + P,(x)o* alm ye Ay athe P(x) eo ® 2), 
where the P;(x) are polynomials, and involves only the rudiments of 
algebraic number theory. 
Dr. Hans HEILBRONN.—On Vinogradov’s solution of Waring’s problem 
(11.15). 
Waring conjectured in the eighteenth century that every positive integer 
can be represented as a sum of a bounded number of k* powers (viz. 
4 squares, 9 cubes, etc.). ‘This was proved by Hilbert in 1909, but his 
method of proof cannot be utilised further in this problem. 
The problem was attacked again by Hardy and Littlewood with the 
most powerful weapons of modern analysis. 
They showed the existence of a number 
G(k) = O(2"R), 
such that every large integer can be represented as sum of at most G (kh) 
positive k*® powers. They also gave an asymptotic formula for the 
number of representations. ‘The most difficult point of their analysis is 
the application of Weyl’s method of diophantine approximations. 
Last year, Vinogradov improved their results to 
G(k) = O(R? log? k) 
by avoiding diophantine approximations altogether, and quite recently 
proved even 
G(k) = O(2 log R), 
a result which comes very near the truth as it can be easily seen that 
G(ki)Z Rk+1. 
Dr. Eric PHILLips.—On the sequence defined by a quadratic recurrence 
formula (11.45). 
1. We investigate the behaviour, as 7 > o, of the sequence { un} defined 
by the recurrence formula 
Un4 y= aUu,? + bu, +c. 
Without loss of generality this can be written 
Unt, — Uy, = (u,, a” ot) (tt, = B). 
Putting « —8 = 2k —1, and writing u, for u, —B —k-+1 we can 
rewrite this again as 
ee 5 th, — (a Bt, te 
Thejbehaviour of the sequence depends on the initial value u, and on the 
value of k. ‘The results are as follows :— 
A. k complex, u, diverges to «©. If k is real we need only consider 
k >, since if k < 4 the réles of k and 1 — & are interchanged. 
B. u,\< —k or > k, u, diverges to , while if u = +k, u, =k for 
alln >1 and ifu,=+(1 —k),u, =1 —kforalln>1; 
C. —k <u, <k,k< k<3,u171 —R; 
D. —k <u, <k, } <k <2, u, oscillates ; 
