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- \( 12^4 = (144)^2 \equiv (144 - 8\cdot17 = 144 - 136 = 8)^2 = 64 \equiv 13
- \( 13^4 = (169)^2 \equiv (16)^2 = 256 \equiv 256 - 15\cdot17 = 256 - 255 = 1 \) â yes
- \( 14^4 = (-3)^4 = 81 \equiv 13
- \( 16^4 = (-1)^4 = 1 \), but \( 16 \equiv -1 \), excluded
- So solutions to \( x^4 \equiv 1 \pmod{17} \) are \( x \equiv 4, 13, 13, \ldots \) â wait: \( 4, 13 \), and also check \( 16 \equiv -1 \): \( (-1)^4 = 1 \), but excluded.
- Wait: \( 4^4 = 256 \equiv 1 \), \( 13^4 = 28561 \), but earlier computations show both â¡ 1.
