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Cunningham numbers
A number    is a Cunningham number if it can be written as    or    for  .

For example,    is a Cunningham number because it is equal to  .

For any fixed base  , the exponents    for which    or    is prime are in general very scarce.

This is due to the fact that    is always divisible by  , and thus it can be prime only if  . Moreover    is always divisible by  , thus a necessary condition for    to be prime is that    is prime as well.

On the other side,    is always divisible by 2 if    is odd, and by  , if    is odd. If    is even because it is of the form    with    odd, then    is divisible by  , hence the only candidates left for primality are of the form    with    even.

In general, the factorization of Cunningham numbers with small bases (and large exponents) has been and is a popular topic in (computational) number theory.

The first Cunningham numbers are 3, 5, 7, 8, 9, 10, 15, 17, 24, 26, 28, 31, 33, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101 more terms

Cunningham numbers can also be... (you may click on names or numbers and on + to get more values)