Combinatorially fruitful properties of 3⋅2−1 and 3⋅2−2 modulo p

Write a ≡ 3 ⋅ 2 − 1 and b ≡ 3 ⋅ 2 − 2 ( mod p ) where p is an odd prime. Let c be a value that is congruent ( mod p ) to either a or b . For any x from Z p ∖ { 0 } , evaluate each of x and c x ( mod p ) within the interval ( 0 , p ) . Then consider the quantity μ c ∗ ( x ) = min ( c x − x , x − c x ) where the differences are evaluated ( mod p − 1 ,  not  mod p ) in the interval ( 0 , p − 1 ) , and the quantity μ c ∧ ( x ) = min ( c x − x , x − c x ) where the differences are evaluated ( mod p + 1 ) in the interval ( 0 , p + 1 ) . As x varies over Z p ∖ { 0 } , the values of each of μ c ∗ ( x ) and μ c ∧ ( x ) give exactly two occurrences of nearly every member of 1 , 2 , … , ( p − 1 ) / 2 . This fact enables a and b to be used in constructing some terraces for Z p − 1 and Z p + 1 from segments of elements that are themselves initially evaluated in Z p .