We must first find the distribution law of wisdom numbers.
Because 2k+ 1 = (k+ 1) 2-k 2,
Obviously, every number greater than 4 and multiples of 4 is also a wise number.
So the even number of 2 divided by 4 is not a wise number.
Therefore, the smallest wisdom number in the natural sequence is 3, and the second wisdom number is 5, starting from 5, followed by 5, 7 and 8; 9, 1 1, 12; 13, 15, 16; 17, 19, 20 ... that is, two odd numbers and multiples of 4 are arranged in groups of three.
3=2^2- 1^2,5=3^2-2^2,7=4^2-3^2,
8=3^2- 1^2,9=5^2-4^2,
1 1=6^2-5^2,……
This is a difficult problem to solve by arithmetic, and it is very simple to solve by algebra.
Let k be a natural number,
∵(k+ 1)^2-k^2=(k+ 1+k)(k+ 1-k)=2k+ 1，
Except 1, all odd numbers are "wisdom numbers".
And: (k+ 1) 2-(k- 1) 2.
=(k+ 1+k- 1)(k+ 1-k+ 1)= 4k，
All even numbers divisible by 4 are "wisdom numbers".
……
If we continue to explore, the problem will be clearer. Natural numbers starting from 1 are divided into four groups. Except for the 1 group, there are three "smart numbers", and the second one in each group is not a "smart number".