Gilbreath's conjecture

Write the primes in order and repeatedly replace the sequence by absolute differences of consecutive terms. The first term of every row is 1.

Gilbreath, of Gilbreath's-shuffle fame in card magic, noticed the pattern in 1958 (Proth had seen it eighty years earlier without publishing). Odlyzko verified it to \(10^{13}\) in 1993, and searches have pushed far further since — yet no proof exists, and no one knows what one would even use: the primes' pseudorandomness suggests truth while defeating every method. A rare open problem a bright schoolchild can start checking by hand.