GCD of a special class of numbers 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., $d &lt; n$ and $d \mid n$), strictly exceeds $1$?

I have checked with the computer several cases and it seems that the GCD is always $&gt; 1$. 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 $&gt; 1$.

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
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