Let $b, n \geq 2$. Is is true that the GCD of all the numbers $(b^n-1)/(b^d-1)$, where $d$ runs over all the proper positive divisors of $n$ (i.e., $d < n$ and $d \mid n$), strictly exceeds $1$?
I have checked with the computer several cases and it seems that the GCD is always $> 1$. It would be helpful if someone could prove/disprove this. Thanks a lot.
Edit: Here's SAGE code and some data for $b = 2$ and $n \in [2, 101]$. The pairs in the output are $(n, GCD)$. The GCD is always $> 1$.
b = 2
for n in [2..101]:
r = b^n
g = (r-1)/(b-1)
div = divisors(n)
for d in div:
if d == n:
continue
g = gcd(g, (r-1)/(b^d-1))
n, g
Output for $b = 2$ and $n \in [2,101].$ The pairs are $(n, GCD)$
(2, 3) (3, 7) (4, 5) (5, 31) (6, 3) (7, 127) (8, 17) (9, 73) (10, 11) (11, 2047) (12, 13) (13, 8191) (14, 43) (15, 151) (16, 257) (17, 131071) (18, 57) (19, 524287) (20, 205) (21, 2359) (22, 683) (23, 8388607) (24, 241) (25, 1082401) (26, 2731) (27, 262657) (28, 3277) (29, 536870911) (30, 331) (31, 2147483647) (32, 65537) (33, 599479) (34, 43691) (35, 8727391) (36, 4033) (37, 137438953471) (38, 174763) (39, 9588151) (40, 61681) (41, 2199023255551) (42, 5419) (43, 8796093022207) (44, 838861) (45, 14709241) (46, 2796203) (47, 140737488355327) (48, 65281) (49, 4432676798593) (50, 1016801) (51, 2454285751) (52, 13421773) (53, 9007199254740991) (54, 261633) (55, 567767102431) (56, 15790321) (57, 39268347319) (58, 178956971) (59, 576460752303423487) (60, 80581) (61, 2305843009213693951) (62, 715827883) (63, 60247241209) (64, 4294967297) (65, 145295143558111) (66, 1397419) (67, 147573952589676412927) (68, 3435973837) (69, 10052678938039) (70, 24214051) (71, 2361183241434822606847) (72, 16773121) (73, 9444732965739290427391) (74, 45812984491) (75, 1065184428001) (76, 54975581389) (77, 581283643249112959) (78, 22366891) (79, 604462909807314587353087) (80, 4278255361) (81, 18014398643699713) (82, 733007751851) (83, 9671406556917033397649407) (84, 20647621) (85, 9520972806333758431) (86, 2932031007403) (87, 41175768098368951) (88, 1034834473201) (89, 618970019642690137449562111) (90, 18837001) (91, 2380065770834284748671) (92, 14073748835533) (93, 658812288653553079) (94, 46912496118443) (95, 2437355091657331538911) (96, 4294901761) (97, 158456325028528675187087900671) (98, 4363953127297) (99, 1010780497307234809) (100, 1098438933505) (101, 2535301200456458802993406410751)
Suppose that not and exist $s,t$ divisors of $n$ satisfying that $\frac{b^n-1}{b^s-1}$ and $ \frac{b^n-1}{b^t-1}$ are coprimes and divisors of $b^n-1$... My solution was wrong