Two functions are defined by Are these functions different?
Details and assumptions: The prime field contains only the numbers and . All computational operations on obey the modular arithmetic with the modulus 5, so that all additions and multiplications in the functions and are always calculated modulo 5. However, the exponents are natural numbers, so that
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
For all primes p and integers x the little theorem of Fermat applies: x p = x ( mod p ) In our case, the modulus p = 5 is also a prime, so we can apply the theorem of Fermat to our prime field. Therefore, in Z / 5 Z applies for x = 0 ⇒ ⇒ x 5 x 3 ⋅ x x − 1 = x = 1 = x 3 In particular, 3 − 1 − 1 = 3 3 = 3 ⋅ ( 9 mod 5 ) = 3 ⋅ 4 = ( 1 2 mod 5 ) = 2 = ( − 1 + 5 ) = 4 Therefore, we can rewrite the function g ( x ) g ( x ) = 4 ⋅ x 5 ⋅ x 2 + 3 − 1 ⋅ x 5 = ( − 1 ) ⋅ x 3 + 2 ⋅ x = f ( x ) Thus, both functions are identical.