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This study explores number theory, proving that a significant proportion of integers can be expressed as the sum of two primes or a prime and a power of two. It also identifies specific arithmetic progressions that cannot be represented in these forms.

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Area of Science:

  • Number Theory
  • Analytic Number Theory

Background:

  • Romanov demonstrated that the set of integers representable as a sum of a prime and a power of two has a positive proportion.
  • Erdős constructed arithmetic progressions of odd integers not expressible as a prime plus a power of two.

Purpose of the Study:

  • To establish results for the proportion of integers of the form p + q^k and p + q, where p and q are primes and k is a positive integer.
  • To investigate arithmetic progressions that exclude integers of these specific forms.
  • To quantify the proportion of integers not representable as p + q^k or p + q.

Main Methods:

  • Utilizing techniques from analytic number theory to establish lower bounds for the density of integers of the specified forms.
  • Constructing specific arithmetic progressions to demonstrate the existence of integers not representable in the studied forms.
  • Applying sieve methods and related tools to estimate the proportion of integers not belonging to these sets.

Main Results:

  • Established that a positive proportion of integers are of the form p + q^k and p + q, where p, q are primes and k is a positive integer.
  • Demonstrated the existence of arithmetic progressions containing no integers of the form p + q^k or p + q.
  • Proved that the proportion of positive integers *not* of the form p + q^k is greater than 1/2, and the proportion *not* of the form p + q is at least 1/3.

Conclusions:

  • The set of integers representable as a prime plus a power of two, or as the sum of two primes, is sparse in a specific sense.
  • There are significant subsets of integers, characterized by arithmetic progressions, that cannot be represented in these additive forms.
  • The study provides precise quantitative estimates for the density of integers that *cannot* be formed by these sums.