Prime number the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems. Prime numbers are also significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures. The evolutionary strategy used by cicadas of the genus "Magicicada" makes use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a
Prime number is a consequence of Fermat's theorem on sums of two squares, which states that an odd prime formula_20 is expressible as the sum of two squares, formula_187, and therefore factorizable as formula_188, exactly when formula_20 is 1 mod 4. Not every ring is a unique factorization domain. For instance, in the ring of numbers formula_190 (for integers formula_34 and formula_35) the number formula_193 has two factorizations formula_194, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion