Congruence of Numbers Raised to Some Power (Part I)

Fermat's Little Theorem states that if p is a prime number, then for any integer n the number n^p − n is an integer multiple of p or as shown in modular notation:

n^p ≡ n (mod p)

or in the simpler form:

n^p−1 ≡ 1 (mod p)

If we modify the expression so that the roles of the exponent and the number raised to some power are reversed, we obtain two equations, one for even and one for odd numbers:

(p−1)²ⁿ ≡ 1 (mod p) (1)
(p−1)²ⁿ⁻¹ ≡ −1 (mod p) (2)

where p can now be either prime or composite. In addition, multiplying both sides of both expressions by p−1 produces:

(p−1)²ⁿ⁺¹ ≡ p−1 (mod p) (3)
(p−1)²ⁿ ≡ −(p−1) (mod p) (4)

It will be shown in Part II that general equations for (1) and (2) have been found as well as four other similar expressions.

Numerical Examples

The following four examples shows the four possible prime/composite modes of mod numbers using 12 as composite and 13 as prime:

I
12⁴ = 20736 ≡ 1(mod 13)
20735/13 = 1595

II
12⁵ = 248832 ≡ −1(mod 13)
248833/13 = 19141

III
11⁴ = 14641 ≡ 1(mod 12)
14640/12 = 1220

IV
11⁵ = 161051 ≡ −1(mod 12)
161052/12 = 13421

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