Note that Every Even perfect numbers (except 6 ) are
 

[LEFT][/LEFT]Note that Every Even perfect numbers (except 6 ) are :


2p−1(2p − 1) = 1 Mod (9*p) but not necessarily alternately.

p ======== perfect NUMBRE

3 ======== 28 =1 mod (27)
5 ======== 496 =1 mod (45)
7 ======== 8128 =1 mod (63)
13 ======== 33550336 =1 mod (117)
17 ======== 8589869056 =1 mod (153)
19 ======== 137438691328 =1 mod (171)
31 ======== 2305843008139952128 =1 mod (279)

is also work as primlity test :



p ======== 2p−1(2p − 1)

15 ======== 536854528 = 28 mod (135)
21 ======== 2199022206976 = 28 mod (189)
35 ======== 590295810341525782528 = 118 mod (315)

If p is prime, then 2[sup]p-1[/sup] == 1 (mod p) and 2[sup]p[/sup] == 2 (mod p) so 2[sup]p-1[/sup](2[sup]p[/sup] - 1) == 1*1 == 1 (mod p).

If p > 3, then p == 1 or 5 (mod 6).

If p == 1 (mod 6) then 2[sup]p-1[/sup] == 1 (mod 9) and 2[sup]p[/sup] - 1 == 1 (mod 9), so 2[sup]p-1[/sup](2[sup]p[/sup] - 1) == 1*1 == 1 (mod 9).

If p == 5 (mod 6) then 2[sup]p-1[/sup] == 7 (mod 9) and 2[sup]p[/sup] - 1 == 4 (mod 9), so 2[sup]p-1[/sup](2[sup]p[/sup] - 1) == 7*4 == 1 (mod 9).

Another triumph for elementary number theory!

lol