Einige Bemerkungen zum Goldbach'schen Problem.
Four prime squares and powers of 2
Four squares of primes and 165 powers of 2
Four squares of primes and powers of 2
By developing the method of Wooley on the quadratic Waring-Goldbach problem, we prove that all sufficiently large even integers can be expressed as a sum of four squares of primes and 46 powers of 2.
Gauss's three squares theorem with almost prime variables
Goldbach conjecture.
Goldbach numbers in sparse sequences
We show that for almost all , the inequalityhas solutions with odd prime numbers and , provided . Moreover, we give a rather sharp bound for the exceptional set.This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.
Goldbach numbers represented by polynomials.
Let N be a large positive real number. It is well known that almost all even integers in the interval [N, 2N] are Goldbach numbers, i.e. a sum of two primes. The same result also holds for short intervals of the form [N, N+H], see Mikawa [4], Perelli-Pintz [7] and Kaczorowski-Perelli-Pintz [3] for the choice of admissible values of H and the size of the exceptional set in several problems in this direction.One may ask if similar results hold for thinner sequences of integers in [N, 2N], of cardinality...
Goldbach’s problem with primes in arithmetic progressions and in short intervals
Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd , Goldbach’s ternary problem is solvable with primes in short intervals with , , and such that has at most prime factors.
Improvement on Davenport's iterative method and new results in additive number theory III
Improvement on Davenport's iterative method and new results in additive number theory, I
Increasing integer sequences and Goldbach's conjecture
Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences.
Integers not of the form
Landau’s problems on primes
At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :(1)Are there infinitely many primes of the form ?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these...
Le crible à vecteurs
Lower and upper bounds for the number of solutions of
Mean values of Dirichlet polynomials and applications to linear equations with prime variables
Montgomery's Weighted Sieve for Dimension Two.
Note sur la question «Tout nombre pair est-il la somme de deux impairs premiers ?»
Numbers representable by five prime squares with primes in an arithmetic progression