We propose a new definition of prime numbers based on the impossibility of decomposing a mosaic of right isosceles triangular tiles into congruent sub-mosaics. This geometric definition is proven to be equivalent to the classical one. As corollaries, we provide a geometric proof of the infinitude of primes, as well as geometric formulations of Goldbach's conjecture, the twin prime problem, and the Collatz conjecture. The proofs are elementary and require only basic geometry and the Pythagorean theorem.