
COMMENTS

Conjecture 1: If m*126 + n = 1729, m*126 > n, then there exists a series with infinitely many Carmichael terms of the form C mod m*234 = n.
(1) For m < 7, we have m*126 < n;
(2) For m = 7, the formula becomes C mod 882 = 847, which includes the Carmichael numbers 1729, 15841, 1033669, etc.;
(3) For m = 8, the formula becomes C mod 1008 = 721, which includes the Carmichael numbers 1729, 15841, 41041, 172081, 670033, 748657, 825265, 997633, etc.;
(4) For m = 9, the formula becomes C mod 1134 = 595, which includes the Carmichael numbers 1729, 1033669, etc.;
(5) For m = 10, the formula becomes C mod 1260 = 469, which includes the Carmichael numbers 1729, 1033669, etc.;
(6) For m = 11, the formula becomes C mod 1386 = 343, which includes the Carmichael numbers 1729, 1082809, etc.;
(7) For m = 12, the formula becomes C mod 1512 = 217, which includes the Carmichael numbers 1729, 41041, etc.;
(8) For m = 13, the formula becomes C mod 1638 = 91, which includes the Carmichael numbers 1729, 41041, 63973, 670033, 997633, etc.
Conjecture 2: If m*234 + n = 1729, m*234 > n, then there exists a series with infinite many Carmichael terms of the form C mod m*234 = n.
(1) For m < 4, we have m*234 < n;
(2) For m = 4, the formula becomes C mod 936 = 793, which includes the Carmichael numbers 1729, 41041, 46657, 126217, 748657, 4909177, 65037817, 193910977, 311388337, 633639097, etc.;
(3) For m = 5, the formula becomes C mod 1170 = 559, which includes the Carmichael numbers 1729, 1033669, 1082809, 7995169, 26921089, etc.;
(4) For m = 6, the formula becomes C mod 1404 = 325, which includes the Carmichael numbers 1729, 41041, 46657, 188461, 314821, etc.;
(5) For m = 7, the formula becomes C mod 1638 = 91, and the case is similar to one from conjecture 1.
Conjecture 3: If m*342 + n = 1729, m*342 > n, then there exists a series with infinitely many Carmichael terms of the form C mod m*342 = n.
(1) For m < 2, we have m*342 < n;
(2) For m = 3, the formula becomes C mod 1026 = 703, which includes the Carmichael numbers 1729, 8911, etc.;
(3) For m = 4, the formula becomes C mod 1368 = 361, which includes the Carmichael numbers: 1729, 126217, etc.;
(4) For m = 5, the formula becomes C mod 1710 = 19, which includes the Carmichael numbers 1729, 1773289, etc.
Conclusion: we can see that 126 = 18*7, 234 = 18*13, and 342 = 18*19, and that 7, 13, and 19 are the prime factors of 1729, so the three conjectures could be expressed all in one. Even more than that, taking randomly another Carmichael number, 8911 = 7*19*67, taking randomly m = 7 in the formula m*18*67, we obtain the formula C mod 8442 = 469, which, indeed, leads to a series of Carmichael numbers: 8911, 1773289, 8830801, etc., which means that the conjecture could be generalized:
Conjecture 4: For any prime factor d of a Carmichael number C1, there exists a series with infinitely many Carmichael terms C2 that satisfy C2 mod (m*18*d) = n, where m*18*d + n = C1, and m and n are natural numbers such that m*18*d < n.
Finally, if we have a Carmichael number divisible by 1729 (e.g., 63973; see the sequence above), we can see that the formula C mod 62244 = 1729 (it can be seen that 62244 + 1729 = 63973) also leads to a series of Carmichael numbers: 126217, etc.; this means that 1729 can be treated like a prime factor. This can probably be generalized to the Carmichael numbers that are divisible by other Carmichael numbers or probably even for a randomly chosen product of prime factors.
