Problems in additive and computational number theory

An old and famous conjecture of Roth, Erdös, Sárko&huml;zy and Sós, is that if you color the numbers [1…N] with colors 1…k, then one color class, must contain two elements i and j such that i + j = x2, for some x. They were able to prove it for k = 3, we will show that the conjecture is true for all k. We will also show that we can replace x2 by any polynomial with integer coefficients f(x) such that 2|f(z) for some z.; Similar to this problem is the problem of determining the upper bound on the density of a set S ⊂ [1…N] such that no two elements i, j ∈ S add up to a square. Erdös conjectured that the density of S is at most ⅓, later Massias exhibited a set of density $1132$ that has this property. Lagarias, Odlyzko and Shearer proved that if a set S is a union of arithmetic progressions, then the density of S can not be bigger than $1132$ if 32|n and ⅓ otherwise. They also showed that for a general set the density of S has to be less than 0.475. Surprisingly for small N (N ≤ 85), using extensive computer search, we can find examples that have more elements than the Massias' example. Nevertheless, we will show that $1132$ is the right density, and prove that for large N the tightest example is the Massias' example.; A famous Conjecture in Mathematics is the Goldbach's Conjecture, which states that, for all even numbers n bigger than 2, n can be written as the sum of two primes. This problem has very long history in number theory, one of the most important results in number theory is Vinogradov's celebrated proof of the ternary Goldbach problem which states that all large enough odd numbers can be written as the sum of three primes. Another approach is by Linnik who showed that the conjecture is essentially true for large n where we write n as a sum of two primes and k powers of 2. Many attempts have been made to reduce the number of powers of 2 in the Linnik's problem. We will show that, using extensive computational power, we can improve on the best published unconditional result by Hongze LI, which states that k = 1906 and show that we can reduce k to 1624. Recently János Pintz was able to make huge improvement in this problem and prove that k could be reduced to 14. We show that our method will lead to an improvement on his result and reduce k to 12.