Forum: Miscellaneous Math
2022-03-23, 13:01
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Replies: 26
Views: 2,335
If n == 3 (mod 8) is prime, then...
If n == 3 (mod 8) is prime, then Mod(Mod(1,n)+x,x^4 + 1)^n == Mod(Mod(1,n) + Mod(1,n)*x^3, x^4 + 1) coefficients of x and x^2 are 0
If n == 5 (mod 8) is prime, then Mod(Mod(1,n)+x,x^4 + 1)^n ==...
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Forum: Miscellaneous Math
2022-03-23, 00:07
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Replies: 26
Views: 2,335
By "your condition" I mean, any of Mod(Mod(1,n) +...
By "your condition" I mean, any of Mod(Mod(1,n) + x, x^4 + 1)^n == Mod(1 + x^r, x^4 + 1) for n == r (mod 8), r =3, 5, or 7.
Any of these conditions imply that Mod(2,n)^n = Mod(2,n).
My...
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Forum: Miscellaneous Math
2022-03-22, 19:43
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Replies: 26
Views: 2,335
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Forum: Miscellaneous Math
2022-03-21, 13:14
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Replies: 26
Views: 2,335
For every integer r = 0 to 7, the sequence an =...
For every integer r = 0 to 7, the sequence an = lift(Mod(x+1,x^4+1)^(8*n + r)), n = 0, 1, 2, ... satisfies the recurrence an+2+136*an+1 + 16*an = 0.
Thus, for any given r, the coefficients of 1,...
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Forum: Miscellaneous Math
2022-03-20, 15:08
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Replies: 26
Views: 2,335
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Forum: Miscellaneous Math
2022-03-18, 15:02
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Replies: 26
Views: 2,335
As you can see from the following, I have...
As you can see from the following, I have obtained closed-form expressions for the coefficients of the degree-less-that-4 lift of Mod(1 + x, 1 + x^4)^n. They can be written as 4-term sums similar to...
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Forum: Miscellaneous Math
2022-03-17, 14:06
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Replies: 26
Views: 2,335
Letv=[Mod(1/4, x^4 - 4*x^3 + 6*x^2 - 4*x + 2),...
Letv=[Mod(1/4, x^4 - 4*x^3 + 6*x^2 - 4*x + 2), Mod(-1/4*x^3 + 3/4*x^2 - 3/4*x + 1/4, x^4 - 4*x^3 + 6*x^2 - 4*x + 2), Mod(-1/4*x^2 + 1/2*x - 1/4, x^4 - 4*x^3 + 6*x^2 - 4*x + 2), Mod(-1/4*x + 1/4, x^4...
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Forum: Miscellaneous Math
2022-03-17, 01:29
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Replies: 26
Views: 2,335
BTW, similar to your observation with...
BTW, similar to your observation with Mod(Mod(1,n)*x + 1, x^4 + 1),
Mod(Mod(1,n)*x + 2, x^2 + 1)^(n+1) == 5 implies 5^(n-1) == 1 (mod n) for n == 3 (mod 4) (assuming 5 does not divide n). That...
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Forum: Miscellaneous Math
2022-03-16, 12:17
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Replies: 26
Views: 2,335
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Forum: Miscellaneous Math
2022-03-16, 01:06
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Replies: 26
Views: 2,335
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Forum: Miscellaneous Math
2022-03-15, 15:16
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Replies: 26
Views: 2,335
(checks, OK, it's equivalent to test proposed...
(checks, OK, it's equivalent to test proposed earlier in thread).
Evidently, none up to 10^8 since you've tested that far. I don't intend to do a numerical sweep above that point, so if you want...
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Forum: Miscellaneous Math
2022-03-15, 14:22
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Replies: 26
Views: 2,335
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Forum: Miscellaneous Math
2022-03-14, 12:36
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Replies: 26
Views: 2,335
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Forum: Miscellaneous Math
2022-03-13, 18:51
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Replies: 26
Views: 2,335
I was able to prove that the indicated...
I was able to prove that the indicated congruences indeed hold if n is a prime congruent to 3, 5, or 7 (mod 8). It made an interesting exercise.
So there are no counterexamples to these with n...
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