with a , b any coprime integers, a > 1 and − a < b < a . (Since a n − b n is always divisible by a − b , the division is necessary for there to be any chance of finding prime numbers. In fact, this number is the same as the Lucas number U n ( a + b , ab ) , since a and b are the roots of the quadratic equation x 2 − ( a + b ) x + ab = 0 , and this number equals 1 when n = 1 ) We can ask which n makes this number prime. It can be shown that such n must be primes themselves or equal to 4, and n can be 4 if and only if a + b = 1 and a 2 + b 2 is prime. (Since a 4 − b 4 / a − b = ( a + b )( a 2 + b 2 ) . Thus, in this case the pair ( a , b ) must be ( x + 1, − x ) and x 2 + ( x + 1) 2 must be prime. That is, x must be in A027861 .) It is a conjecture that for any pair ( a , b ) such that for every natural number r > 1 , a and b are not both perfect r th powers, and −4 ab is not a perfect fourth power . there are infinitely many values of n such that a n − b n / a − b is prime. (When a and b are both perfect r th powers for an r > 1 or when −4 ab is a perfect fourth power, it can be shown that there are at most two n values with this property, since if so, then a n − b n / a − b can be factored algebraically) However, this has not been proved for any single value of ( a , b ) .
As the title suggest, the exhibit theme wrests on the notoriety of Venice as a center of romantic liaisons, seductions and scandals, the likes of Casanova and the many courtesans that frequented the city. Yet, Venice was much more than that. It was a cauldron of intellectual activity, of industry, literature, and music. It was a society that for many years ensured the peace, prosperity and freedom of its people. And it was also a very pious city, one steeped in Roman Catholicism where many churches were built by its richest and most influential citizens.