Eisenstein integers with this property are (*) irreducible.
Samuel Yates coined the term "titanic" for examples of these values that are written in the form of 10 to the power 999 plus n, with the smallest n being equal to 7. These values raised to "p minus 1 divided by 2" are congruent to the Legendre symbol of these values over p mod p according to Euler's criterion. A topological proof about these numbers was given by Hillel Furstenberg. Eisenstein integers with this property are (*) irreducible. Perfect numbers have a one-to-one correspondence with a class of these numbers named for the Frenchman Mersenne. An unproven conjecture states that the number of these with a gap of "two" is infinite. Euclid showed that there are infinitely many of these numbers. For 10 points, name these non-composite numbers which are only divisible by themselves and one.
Samuel Yates coined the term "titanic" for examples of these values that are written in the form of 10 to the power 999 plus n, with the smallest n being equal to 7.
primes [accept any answer mentioning primes] 