>>14112494>>14112535Modified proof now that a mistake has been pointed out
0.
set b = 0 in 1.
1.
set a = 0 in 1.
2.
substitute for each term on left
3.
set b = 1 in 3.
4.
induction
5.
use 2. to substitute in and simplify
6.
consistency check with a = 1 and a = 0
7.
8.
for simplicity, define
9.
10.
then 6. rephrased using new definitions is
11.
plug 11. into 0., rearrange and simplify
12.
Which can only be zero for all integers a and b if 13. and 14. are satisfied
13.
14. (n-2)m = 0
Solving 13. we get
15. or
Solving 14. we get
16. or
Combing the solution set 15. and 16. we get
17. or [/math] n = 0, m = 0 [/text]
So all possible functions f are
18. or
