Some Unconventional Problems in 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 of Erdös’ unconventional problems in number theory, thirty-four years later

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_ k+2 for sufficiently large k? Conway and Guy showed that the answer is affirmative (unpublished) .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

Some Problems in Combinatorial Number Theory

We state and discuss various problems in the general area of arithmetic combinatorics and recent developments related to the ‘sum-product phenomenon’ in the ring of integers, the real and complex numbers and finite fields. In particular, we discuss applications and connections to the theory of exponential sums, Burgess’ estimate, the subspace theorem and to Szemeredi-Trotter type results. In re...