Some Problems in Additive Number Theory

Some Problems in Additive Number Theory

(3) f(x) = (log x/log 2) + 0(1)? 1\Mloser and I asked : Is it true that f(2 11) >_ k+2 for sufficiently large k? Conway and Guy showed that the answer is affirmative (unpublished) . P. Erdös, Problems and results in additive number theory, Colloque, Théorie des Nombres, Bruxelles 1955, p . 137 . 2. Let 1 < a 1< . . . < ak <_ x be a sequence of integers so that all the sums ai,+ . . .+ais, i 1 <...

Sidon in Additive Number Theory . on a Problem of Sidon in Additive Number Theory and on Some Related Problems Addendum

In a note in this Journal [16 (1941), 212-215], Turan and I proved, among other results, the following : Let a l < a2 < . . . < a, < n be a sequence of positive integers such that the sums aj+a; are all different . Then x < n'1 +0(n1 ) . On the other hand, there exist such sequences with x >n1(2---e), for any e >0 . Recently I noticed that J . Singer, in his paper "A theorem in finite projectiv...

Some Recent Developments in Additive Number Theory

Some Problems in Number Theory

A k = max(p,+ t p,), k < p, < p, + , < 2k . In fact I cannot even disprove f(k) = Ak for all sufficiently large k, though it seems likely that f(k) > Ak for all large k. A well known theorem of Pólya and Störmer states that if u > uo(k) then u(u + 1) always contains a prime factor greater than k, thusf(k) can be determined in a finite number of steps, and an explicit bound has been given by Leh...

Some Results on Additive Number Theory'

Let 0 <a 1 <a2< . . . be any infinite sequence of integers . Denote by N(ai , n) the number of ai S n . I conjectured that to every sequence ai there corresponds a sequence b ; of density 0 (i .e ., such that lim n (1/n)N(b;, n)=0) so that every sufficiently large integer is of the form a i +b;. Lorentz 2 in a recent paper proved this conjecture ; in fact, he showed that there exists a sequence...