Finite fields of Mersenne prime order

[toktarev]

Hello.

Here I've described the set of special functions https://mathoverflow.net/questions/3...cial-functions.

It was found of Finite Fields which order is equal to Mersenne's prime order.

For example for GF(31) we have:

2*18 - 1*18=18
3*20 - 2*18=24
4*18 - 3*20=12
5*19 - 4*18=23
6*20 - 5*19=25
7*12 - 6*20=26
8*18 - 7*12=29
9*19 - 8*18=27
10*19 - 9*19=19
11*13 - 10*19=15
12*20 - 11*13=4
13*13 - 12*20=22
14*12 - 13*13=30
15*14 - 14*12=11
16*18 - 15*14=16
17*20 - 16*18=21
18*19 - 17*20=2
19*12 - 18*19=10
20*19 - 19*12=28
21*13 - 20*19=17
22*13 - 21*13=13
23*14 - 22*13=5
24*20 - 23*14=3
25*12 - 24*20=6
26*13 - 25*12=7
27*14 - 26*13=9
28*12 - 27*14=20
29*14 - 28*12=8
30*14 - 29*14=14

As you can see we have involution here, it means:

x*alpha - (x-1)*beta = y
iff
y*delta - (y-1)*beta  = x.

This involution works for any Finite Field of Mersenne's prime order.

Could you please explain why do we have here such involution ?

[toktarev]

x*alpha - (x-1)*beta = y
iff
y*delta - (y-1)*gamma = x.

Fixed