... modulo 29 ,0 ≡ 04,1 ≡ 14≡ 12 4≡ 174≡ 28 4(mod 29 ),7 ≡ 84≡ 94≡ 20 4≡ 21 4(mod 29 ),16 ≡ 2 4≡ 54≡ 24 4≡ 27 4(mod 29 ), 20 ≡ 64≡ 144≡ 154≡ 23 4(mod 29 ), 23 ≡ 34≡ 74≡ 22 4≡ ... ≡ 34≡ 74≡ 22 4≡ 26 4(mod 29 ), 24 ≡ 44≡ 104≡ 194≡ 25 4(mod 29 ), 25 ≡ 114≡ 134≡ 164≡ 184(mod 29 ).The differences −1 − z4are congruent to 28 , 27 , 21 , 12, 8, 5, 4, and 3. None ... 1 (mod 29 ). Then, (xw)4+ (yw)4+ (zw)4is also divisible by 29 . So we canassume that x ≡ 1 (mod 29 ).Thus, we need to show that y4+ z4≡ −1 (mod 29 ), i.e. y4≡ −1 − z4(mod 29 ), is...