Some results in additive number theory I: The critical pair theory

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...

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_ k+2 for sufficiently large k? Conway and Guy showed that the answer is affirmative (unpublished) .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 <...

Some Recent Developments in Additive Number Theory

Some results of number theory

This is a collection of formalized proofs of many results of number theory. The proofs of the Chinese Remainder Theorem and Wilson’s Theorem are due to Rasmussen. The proof of Gauss’s law of quadratic reciprocity is due to Avigad, Gray and Kramer. Proofs can be found in most introductory number theory textbooks; Goldman’s The Queen of Mathematics: a Historically Motivated Guide to Number Theory...

Some Problems and Results in Number Theory

During my very long life I published many papers of similar title . Here I want to discuss some of my favorite problems many of which go back 50 years and which I hope are still alive and will outlive me . Recently Graham and I published a book entitled "Old and new problems and results in combinatorial number theory" Monographic N ° 28 de L'Enseignement Mathématique, Univ . de Genéve . This bo...