Congruence of Numbers Raised to Some Power (Part II)

This is a continuation of Part I where general solutions to (1) and (2) are shown in (5) and (6):

(mp−1)2n ≡ 1 (mod p) ≡ 1 (mod m) ≡ 1 (mod mp)       (5)
(mp−1)2n−1 ≡ −1 (mod p) ≡ 1 (mod m) ≡ 1 (mod mp)       (6)

where p and/or m can be either prime or composite. Note that Part I consists of those m = 1. Four other equations belonging to this group are possible and these are:

(p2r−1)2n ≡ 1 (mod p) ≡ 1 (mod p2r)       (7)
(p2r−1)2n−1 ≡ −1 (mod p) ≡ 1 (mod p2r)       (8)

(p2r−1 −1)2n ≡ 1 (mod p) ≡ 1 (mod p2r−1 )       (9)
(p2r−1−1)2n−1 ≡ −1 (mod p) ≡ −1 (mod p2r−1)       (10)

where r can equal or not equal n. Although all p1 to p2r are included in (7) and (8) and all p1 to p2r−1 are included in (9) and (10) there are cases where the divisors are greater than p2r and p2r−1 as shown in Table I below.

In addition, the p from expressions (7) to (10) may also be multiplied by m to give:

(mpa−1)b ≡ 1 or −1 (mod m) ≡ 1 or −1 (mod p) ≡ 1 or −1 (mod mp) ≡ 1 or −1 (mod mpa)

where a and b take on the exponential values shown.

Numerical Examples

Two examples for (5) and (6) are shown in I and II:

I
(5×7 − 1)2 = 1156 ≡ 1 (mod (5,7,35)
II
(6×7 − 1)3 = 68921 ≡ −1 (mod (6,7,42)

and sometimes other factors. Note the three factors included in the mod() set.

Table I shows the expression to the left of the congruence sign plus their divisors to the the right. The numbers in red correspond to the divisors that are greater than pm in this case p = 2 or 3. x signifies not a divisor.

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Table I
(pm − 1)n ≡ ±1 (mod pm) p1p2p3p4p5p6
(24 − 1)2 ≡ 1 (mod 24) 2 4 8 1632x
(24 − 1)3 ≡ −1 (mod 24) 2 4 8 16xx
(25 − 1)2 ≡ 1 (mod 25) 2 4 8 163264
(25 − 1)3 ≡ −1 (mod 25) 2 4 8 1632x
(34 − 1)2 ≡ 1 (mod 34) 3 9 27 81xx
(34 − 1)3 ≡ −1 (mod 34) 3 9 27 81243x
(35 − 1)2 ≡ 1 (mod 35) 3 9 27 81243x
(35 − 1)3 ≡ −1 (mod 35) 3 9 27 81243729
(44 − 1)2 ≡ 1 (mod 44) 4 16 64 256xx
(44 − 1)3 ≡ −1 (mod 44) 4 16 64 256xx
(45 − 1)2 ≡ 1 (mod 45) 4 16 64 2561024x
(45 − 1)3 ≡ −1 (mod 45) 4 16 64 2561024x
(54 − 1)2 ≡ 1 (mod 54) 5 25 125 625xx
(54 − 1)3 ≡ −1 (mod 54) 5 25 125 625xx
(55 − 1)2 ≡ 1 (mod 55) 5 25 125 6253125x
(55 − 1)3 ≡ −1 (mod 55) 5 25 125 6253125x

Go back to Part I or homepage.

Copyright © 2022 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com