by wasoxygen

There aren't enough small numbers to meet the many demands made of them.

This article is in two parts, the first of which is a do-it-yourself operation, in which I'll show you 35 examples of patterns that seem to appear when we look at several small values of n, in various problems whose answers depend on n. The question will be, in each case: do you think that the pattern persists for all n, or do you believe that it is a figment of the smallness of the values of n that are worked out in the examples?

...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.

16-page PDF

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.