Fermat also conjectured that all numbers of the form 2 2 " n " + 1 are prime ( they are called Fermat numbers ) and he verified this up to " n " = 4 ( or 2 16 + 1 ).
32.
Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime.
33.
In July 1999-while a participant in the Proth Search Group-he became the discoverer of the then-largest known composite Fermat number, a record which his St . Patrick's College ( Drumcondra ) based Proth-Gallot Group twice broke in 2003, the 1999 record having stood until then.
34.
In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers F _ { a } F _ { b } \ dots F _ { s }, a > b > \ dots > s > 1 will be a Fermat pseudoprime to base 2 if and only if 2 ^ s > a.
35.
The double-exponential growth of the Sylvester sequence is unsurprising if one compares it to the sequence of Fermat numbers " F " " n "; the Fermat numbers are usually defined by a doubly exponential formula, 2 ^ { 2 ^ n } + 1, but they can also be defined by a product formula very similar to that defining Sylvester's sequence:
36.
The double-exponential growth of the Sylvester sequence is unsurprising if one compares it to the sequence of Fermat numbers " F " " n "; the Fermat numbers are usually defined by a doubly exponential formula, 2 ^ { 2 ^ n } + 1, but they can also be defined by a product formula very similar to that defining Sylvester's sequence:
37.
Besides, we can define " half generalized Fermat numbers " for an odd base, a half generalized Fermat number to base " a " ( for odd " a " ) is \ frac { a ^ { 2 ^ n } + 1 } { 2 }, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base.
38.
Besides, we can define " half generalized Fermat numbers " for an odd base, a half generalized Fermat number to base " a " ( for odd " a " ) is \ frac { a ^ { 2 ^ n } + 1 } { 2 }, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base.
39.
These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, modulo-2 Pascal's triangle, minus the top row, which corresponds to monogon . ( Because of this, the 1s in such a list form an approximation to the SierpiDski triangle . ) This pattern breaks down after this, as the next Fermat number is composite ( 4294967297 = 641 ?6700417 ), so the following rows do not correspond to constructible polygons.
