Fortunate number Fortunate number A Fortunate number, named after Reo Fortune, is the smallest integer "m" > 1 such that, for a given positive integer "n", "p"# + "m" is a prime number, where the primorial "p"# is the product of the first "n" prime numbers. For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes (2, 3, 5, 7, 11, 13 and 17), which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to
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