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 2019-12-05, 11:15 #1 toktarev   Dec 2019 102 Posts Finite fields of Mersenne prime order 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: Code: 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: Code: 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 ?
2019-12-05, 14:43   #2
toktarev

Dec 2019

216 Posts

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

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