Why is the n-th power of 2 minus 1 divided by 7 equal to N divided by 3?

Why is the n-th power of 2 minus 1 divided by 7 equal to N divided by 3?

Let the remainder of N divided by 3 be m, and discuss the case of M = 0,1,2, M = 1, n = 3K + 1,2 ^ n-1 = 2 ^ (3K + 1) - 1 = 8 ^ k * 2-1, the remainder of N divided by 7 is 1 * 2-1 = 1 (because 8 △ 7 remainder is 1), contradiction! M = 2, n = 3K + 2,2 ^ n-1 = 2 ^ (3K + 2) - 1 = 8 ^ k * 4-1, the remainder of N divided by 7 is 1 * 4-1 = 3 (because 8 △ 7 remainder is 1), contradiction! M

The minimum value of the square + 2x + 2 (x ≥ 0) of the function y = x is?

y=x²+2x+1+1
=(x+1)²+1
So x = 0
Y min = 2

Finding the minimum value of the function y = - Sin ^ 2x-acosx + 2
If the problem, find the process

y=1-sin^2x-acosx+1
=cos^2x-acosx+1
=(cosx-a/2)^2+1-a^2/4
Because cosx ∈ [- 1,1]
therefore
(1) When a / 2 > 1, that is, when a > 2
When cosx = 1, Ymin = 2-A
（2） a/2