A. M. Samoilenkos method for the determination of the by Trofimchuk E. P.

By Trofimchuk E. P.

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Let us turn to a flourishing area of mathematics that was practically created by Erd˝ os. This is the theory of random graphs, started by Erd˝ os and then, a little later, founded by Erd˝ os and R´enyi, Throughout his career, Erd˝os had a keen eye for problems likely to yield to either combinatorial or probabilistic attacks. Thus it is not surprising that he had such a tremendous success in combining combinatorics and probability. At first, Erd˝ os used random methods to tackle problems in mainstream graph theory.

Independently, Erd˝ os and Ricci showed that the set of limit points of the sequence (pn+1 − pn )/ log pn has positive Lebesgue density and yet, no number is known to be a limit point. Concerning large gaps between consecutive primes, Backlund proved in 1929 that lim supn→∞ (pn+1 − pn )/ log pn ≥ 2. In quick succession, this was improved by Brauer and Zeitz (1930), by Westzynthius (1931), and then by Ricci (1934), to lim sup n→∞ pn+1 −pn > 0. log pn log log log pn By making use of the method of Brauer and Zeitz, Erd˝ os proved in 1934 that lim sup n→∞ (pn+1 −pn )(log log log pn )2 > 0.

Call S an asymptotic basis of order k if rk (n) ≥ 1 whenever n is sufficiently large. In 1932 Sidon asked Erd˝ os the following question. Is there an asymptotic basis of order 2 such that r2 (n) = o(n ) for every > 0? The young Erd˝os confidently reassured Sidon that he would come up with such a sequence. Erd˝os was right, but it took him over 20 years: he proved in 1954 in Acta (Szeged) that for some constant c there is a sequence S such that 1 ≤ r2 (n) < c log n if n is large enough. What can one say about rk (n) rather than r2 (n)?

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