Abstract
On pairs of equations involving unlike powers of primes and powers of 2
Highlights
In the 1950s, Linnik [2, 3] proved that each large even integer N is a sum of two primes and a bounded number of powers of 2, N = p1 + p2 + 2v1 + 2v2 + · · · + 2vk1, (1)
The famous Goldbach conjecture implies that k1 = 0
The explicit value for the number k1 was improved by many authors
Summary
In the 1950s, Linnik [2, 3] proved that each large even integer N is a sum of two primes and a bounded number of powers of 2,. In 2001, Liu and Liu [4] proved that every large even integer N can be written as a sum of eight cubes of primes and k3 powers of 2, p23. Trudgian [10] improved the value of k to 156, to 16 by Zhao [12] and 15 by Lü [9]. We obtain a further improvement of the value of k by giving the following theorem. For k = 302, the equations (5) are solvable for every pair of sufficiently large positive even integers N1 and N2 satisfying N2 N1 > N2
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