On a diophantine problem with one prime, two squares of primes and s powers of two
Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality has infinitely many solutions in prime variables p₁, p₂, p₃.
We prove that almost all positive even integers can be represented as with for . As a consequence, we show that each sufficiently large odd integer can be written as with for .
Let be the polynomial ring over the finite field , and let be the subset of containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that .