Additive and Multiplicative Properties of Prime Numbers

Prime numbers lie at the heart of number theory through two complementary lenses: additive structure, which explores ways in which primes combine in sums and Diophantine relations, and multiplicative structure, which concerns their distribution in products, arithmetic progressions and sparse sequences. On the additive side, classical questions such as Goldbach’s conjecture, sums of prime powers and Waring–Goldbach problems have spurred intricate developments in exponential‐sum estimates, sieve methods and mean-value theorems. Multiplicative investigations probe the spacing of primes, the behaviour of multiplicative functions on prime arguments, and the existence of primes in highly regular or extremely thin sets that still satisfy global laws such as the prime number theorem. Modern research weaves these strands, deploying tools from harmonic analysis, probabilistic models and computational verification to reveal deep regularities and unexpected irregularities in both sum‐based representations and multiplicative patterns of primes.

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Additive and Multiplicative Properties of Prime Numbers publication trend

The graph below shows the total number of articles in additive and multiplicative properties of prime numbers across all publications each year (not limited to Nature Index journals).

Technical terms

Diophantine inequality: An inequality involving integer or prime variables seeking solutions within a real or integer range.

Prime number theorem: A foundational result describing the asymptotic density of primes among the integers, typically π(x) ∼ x / log x as x → ∞.

Waring–Goldbach problem: A hybrid of Waring’s problem for sums of k-th powers and Goldbach‐type representations, asking which integers can be expressed as sums of prime powers.

Asymptotic formula: An expression that describes the leading behaviour of a counting or sum function in the limit of large argument, often including an explicit main term and an error estimate.

References

1. Goldbach–Linnik–Type Problem of Symmetric Mixed Powers of Primes and Powers of Two. Symmetry (2024).
2. Sparse sets that satisfy the prime number theorem. Journal of Number Theory (2024).
3. On a binary Diophantine inequality involving prime numbers. AIMS Mathematics (2024).