On some Waring–Goldbach problems

Key Points

• The research aims to explore how sufficiently large odd integers can be expressed as sums of different powers of primes.
• Establishing mathematical proofs for representations of odd integers
• Analyzing sums involving squares, cubes, and higher powers of primes
• Addressing divisibility conditions related to the number 3
• Every sufficiently large odd integer can be written as a sum of two squares, a cube, a fourth power, a fifth power, and two sixth powers of primes.
• Every sufficiently large odd integer not divisible by 3 can be expressed as a sum of two squares, three fourth powers, a fifth power, and a sixth power of primes.