Flavors of the Green-Tao theorem

The Green-Tao theorem states that prime numbers contain infinitely long arithmetic progressions—sequences where each term increases by a fixed amount. The "flavors" of this theorem refer to various extensions and related results, such as discovering similar patterns within other types of numbers or under different conditions. These variations explore how prime-like structures appear in broader mathematical contexts, revealing deep connections between prime distribution, combinatorics, and number theory. Overall, these flavors demonstrate that special, orderly patterns are more common in the prime numbers than one might initially expect.