There aren't enough small numbers to meet the many demands made of them.
...This first part contains no information; rather it contains a good deal of disinformation. The first part contains one theorem: You can't tell by looking. It has wide application, outside mathematics as well as within. It will be proved by intimidation.
Example 1. 2^(2 ^ 0) + 1 = 3 2^(2 ^ 1) + 1 = 5
2^(2 ^ 2) + 1 = 17
2^(2 ^ 3) + 1 = 257
2^(2 ^ 4) + 1 = 65537
The sums are all primes.
Example 15. (x+y)^3 = x^3 + y^3 + 3xy(x+y)(X^2 + xy + y^2)^0 (x+y)^5 = x^5 + y^5 + 5xy(x+y)(X^2 + xy + y^2)^1
(x+y)^7 = x^7 + y^7 + 7xy(x+y)(X^2 + xy + y^2)^2
Example 16.The sequence of centered hexagonal numbers begins 1, 7, 19, 37, 61, ...
The partial sums of this sequence, 1, 8, 27, 64, 125, appear to be perfect cubes.
Well this isn't fair because everybody already knows that there exists no known set of functions, mappings, transformations etc. that generates prime numbers.
What kills me is when mathematicians turn to wordsmithing:
Capricious coincidences cause careless conjectures.
Early exceptions eclipse eventual essentials.
Initial irregularities inhibit incisive intuition.
And when I say "kills me", I secretly love it.