  The prime numbers have been a source of fascination for millenia and continue
to surprise us. Motivated by the hyperuniformity concept, which has attracted
recent attention in physics and materials science, we show that the prime
numbers in certain large intervals possess unanticipated order across length
scales and represent the first example of a new class of many-particle systems
with pure point diffraction patterns, which we call {\it effectively
limit-periodic}. In particular, the primes in this regime are hyperuniform.
This is shown analytically using the structure factor $S(k)$, proportional to
the scattering intensity from a many-particle system. Remarkably, the structure
factor for primes is characterized by dense Bragg peaks, like a quasicrystal,
but positioned at certain rational wavenumbers, like a limit-periodic point
pattern. We identify a transition between ordered and disordered prime regimes
that depends on the intervals studied. Our analysis leads to an algorithm that
enables one to predict primes with high accuracy. Effective limit-periodicity
deserves future investigation in physics, independent of its link to the
primes.
